Week 3 · Chapter

Chapter 3 — Functions: Notation, Domain & Range

College Algebra Free sample edition

The module's primary reading · ~40–50 minutes · Objective 3 (use function notation; determine the domain and range of a function; evaluate functions at numeric and algebraic inputs; and form sums, differences, products, quotients, and compositions of functions) · Read it first this week; the lecture, the tutorial, the quiz, and the Model/Build Studio all build on it.

In plain language

Press button 4 on a snack machine and one bag drops. Tell a bowl shop you want 3 toppings and the register shows one price, 13 dollars. This week you learn the math name for that kind of rule, a function (a rule that gives exactly one answer for each thing you put in). You will learn a short way to write a function, how to plug numbers and expressions into it, and how to tell which inputs it can accept, called its domain (the list of inputs that work). You will also learn its range (the list of answers it can give back), and how to chain two functions together so that one feeds the next.

Why you already care about this

A bowl shop charges 4 dollars for a plain bowl and 3 dollars for each topping. With 2 toppings you pay 10 dollars, and with 5 toppings you pay 19 dollars. With 3 toppings the register shows 13 dollars, every single time: one topping count, one price. You have been living with rules like this for years, every time a price, a grade, or a fare depended on one number you chose.

Rules like this are everywhere, and so are their limits. A shop that sells 0 to 5 toppings cannot sell you −2 toppings or 7.5 toppings. A calculator flashes an error if you ask for the square root of a negative number or divide by zero. Knowing which inputs a rule accepts is half of using it well. This chapter gives you the words, the notation, and the checks to handle any rule like this with confidence.

By the end of this chapter you will be able to:

decide whether a mapping, a table, or a graph is a function, and use the vertical line test;
evaluate a function at numbers, including negative numbers, and at expressions such as a + 2;
find the domain of a polynomial, a rational function, and a square-root function, and write it in interval notation;
read a range from a graph or a table, including in a real setting;
add, subtract, multiply, divide, and compose two functions, and explain why the order of a composition matters.

Carry this sentence with you: parentheses first, inside first — never guess the order.

Where we are

Already covered: Week 1, Real Numbers, Exponents & Algebraic Expressions, taught you to evaluate an expression with the order of operations and to handle a negative base. Week 2, Linear Equations & Inequalities, taught you to solve for an unknown and to write a set of numbers in interval notation.
This week: functions: what makes a rule a function, function notation, evaluating at numbers and at expressions, domain and range, combining two functions, and composition.
How they connect: evaluating a function is Week 1's evaluation with a new name, and a domain is a solution set from Week 2, written in the same interval notation.
Where we're going next: Week 4, Linear Functions & Their Graphs, takes the simplest function of all, the straight line, and reads its slope and its starting value.

Two earlier skills carry this week. From Week 1, a negative base needs parentheses before you square it: (−2)² = (−2)(−2) = 4, while −2² = −4. From Week 2, interval notation uses a square bracket for an endpoint you include and a parenthesis for one you leave out, and infinity always gets a parenthesis. So [5, ∞) means "5 and every number above it", and (−∞, 3) means "every number below 3". You will use both moves in nearly every section below.

1. What a function is: one input, one output

Think about the bowl shop again. You tell the cashier a number of toppings, and the register gives back one price. You would be upset if the same order rang up as 10 dollars one minute and 13 dollars the next. That "one answer every time" promise is what makes the price rule a function.

A function is a rule that pairs each input with exactly one output; two different inputs may share an output, but no input may have two outputs.
An input is a value you feed into a function, usually written x; it is the first number in each input-output pair.
An output is the value a function returns for a given input, written f(x) for the input x; it is the second number in each input-output pair.
A relation is any set of input-output pairs, with no limit on how many outputs one input may have; a function is the special kind of relation in which every input has exactly one output.

A function can show up in three forms. A mapping lists inputs on the left and draws an arrow from each input to its output. A table puts inputs in one column and outputs in the next. A graph plots each pair as a point.

Worked example — the bowl shop as a mapping. The shop's price for x toppings is 3 dollars per topping plus the 4-dollar bowl. Work out each price.

0 toppings: 3 · 0 + 4 = 0 + 4 = 4 dollars.
1 topping: 3 · 1 + 4 = 3 + 4 = 7 dollars.
2 toppings: 3 · 2 + 4 = 6 + 4 = 10 dollars.
3 toppings: 3 · 3 + 4 = 9 + 4 = 13 dollars.
4 toppings: 3 · 4 + 4 = 12 + 4 = 16 dollars.
5 toppings: 3 · 5 + 4 = 15 + 4 = 19 dollars.

So the mapping sends 0 → 4, 1 → 7, 2 → 10, 3 → 13, 4 → 16, and 5 → 19. Each topping count has exactly one arrow leaving it, so the rule is a function.

A table works the same way. The table below follows a 100-gram sample that loses half its mass each period.

Mass of a 100-gram sample after each period, in grams
Periods passed	Mass left (grams)
0	100
1	50
2	25
3	12.5
4	6.25

Each period count sits beside exactly one mass, so this table is a function too. In words: tell me how many periods have passed, and I can tell you one mass.

Here is the part that surprises people. Two inputs are allowed to share an output. Take the squaring rule p(x) = x². Then p(−2) = (−2)² = 4 and p(2) = 2² = 4. Two inputs, −2 and 2, both land on 4. That is fine. The rule still gives each input one answer.

Misconception check — sharing an output is not breaking the rule. Students often think a function cannot repeat an output, so p(x) = x² "fails" because −2 and 2 both give 4. The rule is about inputs, not outputs. A function breaks only when one input has two outputs. Two people can share a birthday; one person cannot have two birthdays.

The vertical line test: check a graph by eye

Picture a graph drawn on graph paper. Every vertical line on the paper stands for one input, one value of x. If a vertical line touches the graph at two points, that one input has two outputs. So a graph shows a function exactly when no vertical line touches it more than once.

The vertical line test is a graph check for whether a graph shows y as a function of x: if any vertical line meets the graph more than once, the graph is not a function.

The graph of y = x² is a U-shaped curve, a parabola, that opens upward with its lowest point at (0, 0). Slide a vertical line from left to right across it. Wherever you stop, the line crosses the curve exactly once. The parabola passes.

Worked example — a sideways curve. The graph of x = y² is the same U shape turned on its side, opening to the right. Test the input x = 4.

Ask which y-values pair with x = 4. You need y² = 4.
Try y = 2. 2² = 2 · 2 = 4, so (4, 2) is on the curve.
Try y = −2. (−2)² = (−2)(−2) = 4, so (4, −2) is on the curve too.
Read the result. The vertical line x = 4 meets the curve at two points, (4, 2) and (4, −2).

The input 4 has two outputs, so x = y² is a relation but not a function of x.

In practice (your tools). In Desmos, type y = x^2 on one line and x = y^2 on the next. Then type x = 4 on a third line. The vertical line crosses the first curve once and the second curve twice. You can drag a slider on x = a to sweep the line across both graphs.

Misconception check — a horizontal line is the wrong test. Students often slide a horizontal line instead, see it hit the parabola y = x² twice, and call the parabola "not a function". A horizontal line holds an output fixed, so two hits only mean two inputs share that output, which the start of this section showed is allowed. The test is vertical because each vertical line is one input.

2. Function notation: one name, one output

When a rule has a name, you can talk about it quickly. Instead of saying "the output of the rule 3x² − 2x + 5 when the input is −2", you write f(−2). The letter f names the function. The input goes inside the parentheses.

Function notation is the way of writing a function's output as f(x), read "f of x", where f names the function and x is the input; f(x) is one output value, not f multiplied by x.

This week's main function is f(x) = 3x² − 2x + 5. To evaluate it, replace every x with the input, wrapped in parentheses. The parentheses matter most when the input is negative.

Worked example — evaluating f at three inputs. Feed in −2, then 0, then 3, one step at a time.

f(−2). Replace each x with (−2): 3(−2)² − 2(−2) + 5. Square first: (−2)² = 4. Multiply: 3 · 4 = 12, and −2(−2) = +4. Add: 12 + 4 + 5 = 21. So f(−2) = 21.
f(0). 3 · 0² − 2 · 0 + 5 = 0 − 0 + 5 = 5. So f(0) = 5.
f(3). 3 · 3² − 2 · 3 + 5 = 3 · 9 − 6 + 5 = 27 − 6 + 5 = 26. So f(3) = 26.

In words: feed in −2 and the function returns 21; feed in 0 and it returns 5; feed in 3 and it returns 26.

The same rule at all six whole-number inputs from −2 to 3 makes a function table. These six values are the week's shared table.

Inputs and outputs of the function f at six whole-number inputs
x	−2	−1	0	1	2	3
f(x)	21	10	5	6	13	26

The six outputs add to 81. The smallest is 5, at x = 0, and the largest is 26, at x = 3.

In practice (your tools). In Desmos, type f(x) = 3x^2 - 2x + 5 on one line. On the next line type f(-2), and Desmos shows 21. Try f(0) and f(3) and check that you get 5 and 26.

Misconception check — the square belongs to the whole negative number. Students often write 3(−2)² as 3(−4) and get −12 + 4 + 5 = −3. It is tempting because the minus sign looks like it comes along for the ride. But the parentheses say square the whole number −2, and (−2)(−2) = +4. So the first term is +12, not −12, and f(−2) = 21. Wrap the input in parentheses before you do anything else.

A second slip is to read f(x) as "f times x". It never means that. If f(3) meant f · 3, there would be nothing to multiply, because f is a name, not a number.

3. Algebraic inputs: substitute the whole expression

Go back to the bowl shop. Suppose a friend orders 2 more toppings than you do, and you call your topping count a. Your friend's price is the shop's rule with a + 2 fed in. The input is now an expression, not a single number, but the rule does not change.

An algebraic input is an input that is an expression with a letter in it, such as a + 2 or −x, rather than a single number; you substitute the whole expression wherever the variable appears.

Name the bowl shop's rule g(x) = 3x + 4. The letter a stands for any number.

Worked example — the friend's bowl, g(a + 2). Feed the expression a + 2 into g.

Substitute the whole input. Replace x with (a + 2): g(a + 2) = 3(a + 2) + 4.
Distribute the 3 to both terms. 3(a + 2) = 3a + 6.
Collect the numbers. 3a + 6 + 4 = 3a + 10.

So g(a + 2) = 3a + 10. Check it with a number. If you order a = 1 topping, your friend has 3 toppings, and the shop charges 3 · 3 + 4 = 13 dollars. The formula gives 3 · 1 + 10 = 13. The two agree.

The same move handles other inputs:

g(−x) = 3(−x) + 4 = −3x + 4.
g(x + h) = 3(x + h) + 4 = 3x + 3h + 4. Here h is a number added to the input, not a new function.
f(−x) = 3(−x)² − 2(−x) + 5 = 3x² + 2x + 5, because (−x)² = x² and −2(−x) = +2x.

Misconception check — distribute to every term. Students often write g(a + 2) = 3a + 2 + 4 = 3a + 6. It is tempting to multiply only the first term in the parentheses. Check it with a = 1: the wrong formula gives 3 + 6 = 9 dollars, but 3 toppings really cost 13 dollars. The 3 multiplies the whole input, so it multiplies the 2 as well.

Pause and reflect — choose one. Pick either of the 2 prompts; they count the same.

(a) In your own words, explain to a friend why g(a + 2) is not the same as g(a) + 2. Use the bowl shop if it helps.
(b) Think of a rule from your own life where you feed in a number and get one result, such as a fare or a pay rate. Describe what feeding in "2 more" would mean for that rule.

4. Domain: which inputs does the rule accept?

A calculator gives up on two kinds of requests: dividing by zero, and taking the square root of a negative number. Any input that forces one of those is an input the rule cannot use. Finding the domain means finding every input the rule can use.

The domain is the set of every input a function can accept and still give a real-number output, usually written in interval notation.

Three rules cover this week's functions. Each one is a property of the kind of formula you have.

A polynomial (a sum of terms made of numbers and whole-number powers of x, like 3x² − 2x + 5) accepts every real number. Its domain is (−∞, ∞).
A rational function leaves out the inputs that make its denominator zero.
An even-root radical, like a square root, needs the expression under the root to be zero or more.

A rational function is a function written as one polynomial divided by another; any input that makes the denominator zero is left out of its domain.
The radicand is the expression under a radical sign, such as the x − 5 inside the square root of x − 5; for a square root it must be zero or more to give a real output.

Worked example — two domains. Find the inputs each rule can accept.

The rational function r(x) = (x + 1) ⁄ (x − 3). Set the denominator equal to zero: x − 3 = 0, so x = 3. At x = 3 the rule asks for (3 + 1) ⁄ (3 − 3) = 4 ⁄ 0, which is undefined. Every other input works. The domain is (−∞, 3) ∪ (3, ∞), every real number except 3.
The square-root function s(x) = √(x − 5). The radicand must be zero or more: x − 5 ≥ 0. Add 5 to both sides: x ≥ 5. The domain is [5, ∞). Check the endpoint: s(5) = √(5 − 5) = √0 = 0, a real number, so 5 belongs. Check a number below it: s(4) = √(4 − 5) = √(−1), which is not a real number, so 4 does not belong.

A real setting can shrink a domain further.

A practical domain is the inputs that make sense in a real setting, which can be fewer than the formula allows, such as only whole numbers of toppings from 0 to 5.

The formula g(x) = 3x + 4 accepts every real number. The bowl shop does not. It sells whole toppings from 0 to 5, so its practical domain is {0, 1, 2, 3, 4, 5}. In the same way, a launched object's height only makes sense from the launch until it lands.

Misconception check — the endpoint of a square root is included. Students often write the domain of √(x − 5) as x > 5, leaving out 5. It feels as if the edge must be off-limits, the way 3 is off-limits for r. But √0 = 0 is a perfectly good output. A zero under a square root is fine; a zero in a denominator is not. So the domain is [5, ∞), with a square bracket.

Honesty note: other courses sometimes write a domain in set-builder form, {x | x ≥ 5}. That says the same thing. In this course you write interval notation.

5. Range: what comes out?

The domain asks what can go in. The range asks what actually comes out. For the bowl shop, the prices that come out are 4, 7, 10, 13, 16, and 19 dollars, so its range in that setting is {4, 7, 10, 13, 16, 19}.

The range is the set of every output a function actually produces as its input runs through the domain.

From a graph, you read the range by looking up and down: find the lowest output the graph reaches and the highest. For p(x) = x², the U-shaped graph bottoms out at 0 and climbs forever, and a square is never negative. Its range is [0, ∞). The graph of s(x) = √(x − 5) starts at the point (5, 0) and rises slowly to the right without end. Its range is also [0, ∞).

Worked example — the launched object. An object is launched upward from an 80-foot platform. Its height in feet after t seconds is h(t) = −16t² + 64t + 80. The table shows its height each second until it lands.

Height of the launched object in feet at each whole second from launch to landing
t (seconds)	0	1	2	3	4	5
h(t) (feet)	80	128	144	128	80	0

Set the practical domain. The flight runs from launch, t = 0, to landing, t = 5, so the domain is [0, 5].
Find the highest output. The heights rise from 80 to 128 to 144, then fall. The top of the flight is 144 feet, at t = 2.
Find the lowest output. The object lands at h(5) = 0 feet. Check: −16(25) + 64(5) + 80 = −400 + 320 + 80 = 0.
Write the range. Every height from 0 up to 144 occurs during the flight, so the range is [0, 144].

In words: during its flight, the object is never lower than the ground and never higher than 144 feet.

Caveat: this height model ignores air resistance. It is the standard simplified model, good for learning, not a promise about a real launch on a windy day.

In practice (your tools). In Desmos, type h(x) = -16x^2 + 64x + 80 {0 <= x <= 5}. The curly-brace part limits the graph to the domain [0, 5]. Click the top of the curve to see the point (2, 144).

Misconception check — the range is outputs, not inputs. Students often answer a range question with the domain, writing [0, 5] for the launched object because those numbers sit on the horizontal axis. The range lives on the vertical axis. Ask "how high and how low does the graph go?", not "how far left and right?".

6. Combine two functions: add, subtract, multiply, divide

Suppose one rule gives a number and another rule gives a second number for the same input. You can add those two numbers, subtract them, multiply them, or divide them. Doing that for every input makes a new function.

Operations on functions means making a new function by adding, subtracting, multiplying, or dividing the outputs of two functions at the same input; a quotient leaves out every input where the bottom function equals zero.

Use the week's two named functions, f(x) = 3x² − 2x + 5 and g(x) = 3x + 4.

Worked example — four new functions. Combine f and g at the same input four ways.

Sum. (f + g)(x) = (3x² − 2x + 5) + (3x + 4). Collect like terms: −2x + 3x = x and 5 + 4 = 9. So (f + g)(x) = 3x² + x + 9. Check at x = 2: f(2) + g(2) = 13 + 10 = 23, and 3 · 4 + 2 + 9 = 23.
Difference. (f − g)(x) = (3x² − 2x + 5) − (3x + 4). Distribute the minus to both terms: 3x² − 2x + 5 − 3x − 4. Collect: −2x − 3x = −5x and 5 − 4 = 1. So (f − g)(x) = 3x² − 5x + 1. Check at x = 2: 13 − 10 = 3, and 3 · 4 − 10 + 1 = 3.
Product at a point. (f · g)(1) = f(1) · g(1). Find each: f(1) = 3 − 2 + 5 = 6 and g(1) = 3 + 4 = 7. Multiply: 6 · 7 = 42.
Quotient. (f ⁄ g)(x) = (3x² − 2x + 5) ⁄ (3x + 4). The bottom is zero when 3x + 4 = 0, that is, 3x = −4 and x = −4 ⁄ 3. So x ≠ −4 ⁄ 3. At x = 2: (f ⁄ g)(2) = 13 ⁄ 10 = 1.3.

You will multiply two polynomials into one long formula in Week 6. For now, a product at a single point is enough.

Misconception check — the minus sign reaches every term. Students often subtract only the first term of g and write 3x² − 2x + 5 − 3x + 4 = 3x² − 5x + 9. It is tempting because the parentheses disappear. Test it at x = 2: the wrong formula gives 12 − 10 + 9 = 11, but f(2) − g(2) = 13 − 10 = 3. The minus sign belongs to the whole of g, so +4 becomes −4.

7. Composition: the inside goes first

At the bowl shop's register, a student discount takes 2 dollars off the price. Call that rule d(y) = y − 2, where y is a price. To get your final cost, first find the price, then apply the discount. One function's output becomes the next function's input. That chaining is called composition.

Composition is using one function's output as another function's input; the composition of f and g, written f(g(x)) and read "f of g of x", applies g first and then applies f to the result.
The inner function is, in a composition, the function applied first; in f(g(x)) the inner function is g.
The outer function is, in a composition, the function applied second, to the inner function's output; in f(g(x)) the outer function is f.

The short symbol for composition is a small circle: (f ∘ g)(x) = f(g(x)). Read the symbol from the inside out.

With 3 toppings, the price is g(3) = 3 · 3 + 4 = 13 dollars, and the discount gives d(13) = 13 − 2 = 11 dollars. That is (d ∘ g)(3) = 11. Now flip the order: (g ∘ d)(3) = g(d(3)) = g(1) = 3 · 1 + 4 = 7. That would mean taking 2 off the topping count before pricing, which is a different deal. Order matters.

Worked example — the small pair p and q. Let p(x) = x² and q(x) = x + 1.

(p ∘ q)(2): inside first. q(2) = 2 + 1 = 3. Then p(3) = 3² = 9. So (p ∘ q)(2) = 9.
(q ∘ p)(2): now p is inside. p(2) = 2² = 4. Then q(4) = 4 + 1 = 5. So (q ∘ p)(2) = 5.
Compare with the product. p(2) · q(2) = 4 · 3 = 12.

Three different numbers: 9, 5, and 12. As formulas, (p ∘ q)(x) = (x + 1)² and (q ∘ p)(x) = x² + 1. Check (q ∘ p)(2) = 2² + 1 = 5.

Worked example — the week's named functions. Now compose f and g at the input −2.

(f ∘ g)(−2). Inside first: g(−2) = 3(−2) + 4 = −6 + 4 = −2. Then f(−2) = 21. So (f ∘ g)(−2) = 21.
(g ∘ f)(−2). Inside first: f(−2) = 21. Then g(21) = 3 · 21 + 4 = 63 + 4 = 67. So (g ∘ f)(−2) = 67.
A formula for g ∘ f. Feed all of f into g: (g ∘ f)(x) = 3(3x² − 2x + 5) + 4. Distribute: 9x² − 6x + 15 + 4 = 9x² − 6x + 19. Check at x = −2: 9 · 4 + 12 + 19 = 36 + 12 + 19 = 67.

Same two functions, same input, and the answers are 21 versus 67, about a 3.2-fold difference. Order is not a detail.

In practice (your tools). In Desmos, with f(x) = 3x^2 - 2x + 5 and g(x) = 3x + 4 already typed, enter f(g(-2)) and then g(f(-2)). Desmos shows 21 and 67.

Misconception check — a composition is not a product, and it runs inside out. Students often read (f ∘ g)(x) as f(x) · g(x), or apply f first because f is written first. Both slips are tempting because the symbols sit side by side. For the small pair at 2, the product is 12, applying p first gives 5, and the true (p ∘ q)(2) is 9. The cure: rewrite (f ∘ g)(x) as f(g(x)), then work from the innermost parentheses out.

Pause and reflect — choose one. Pick either of the 2 prompts; they count the same.

(a) Describe a two-step process from your own day where the order of the steps changes the result, such as a discount and a tax, and say which step is the inner function.
(b) Explain in your own words why (f ∘ g)(x) is read from the right, even though f is written on the left.

Key terms

function — A rule that pairs each input with exactly one output; two different inputs may share an output, but no input may have two outputs. (first used: 1. What a function is: one input, one output)
input — A value you feed into a function, usually written x; it is the first number in each input-output pair. (first used: 1. What a function is: one input, one output)
output — The value a function returns for a given input, written f(x) for the input x; it is the second number in each input-output pair. (first used: 1. What a function is: one input, one output)
relation — Any set of input-output pairs, with no limit on how many outputs one input may have; a function is the special kind of relation in which every input has exactly one output. (first used: 1. What a function is: one input, one output)
vertical line test — A graph check for whether a graph shows y as a function of x: if any vertical line meets the graph more than once, the graph is not a function. (first used: 1. What a function is: one input, one output)
function notation — The way of writing a function's output as f(x), read "f of x", where f names the function and x is the input; f(x) is one output value, not f multiplied by x. (first used: 2. Function notation: one name, one output)
algebraic input — An input that is an expression with a letter in it, such as a + 2 or −x, rather than a single number; you substitute the whole expression wherever the variable appears. (first used: 3. Algebraic inputs: substitute the whole expression)
domain — The set of every input a function can accept and still give a real-number output, usually written in interval notation. (first used: 4. Domain: which inputs does the rule accept?)
rational function — A function written as one polynomial divided by another; any input that makes the denominator zero is left out of its domain. (first used: 4. Domain: which inputs does the rule accept?)
radicand — The expression under a radical sign, such as the x − 5 inside the square root of x − 5; for a square root it must be zero or more to give a real output. (first used: 4. Domain: which inputs does the rule accept?)
practical domain — The inputs that make sense in a real setting, which can be fewer than the formula allows, such as only whole numbers of toppings from 0 to 5. (first used: 4. Domain: which inputs does the rule accept?)
range — The set of every output a function actually produces as its input runs through the domain. (first used: 5. Range: what comes out?)
operations on functions — Making a new function by adding, subtracting, multiplying, or dividing the outputs of two functions at the same input; a quotient leaves out every input where the bottom function equals zero. (first used: 6. Combine two functions: add, subtract, multiply, divide)
composition — Using one function's output as another function's input; the composition of f and g, written f(g(x)) and read "f of g of x", applies g first and then applies f to the result. (first used: 7. Composition: the inside goes first)
inner function — In a composition, the function applied first; in f(g(x)) the inner function is g. (first used: 7. Composition: the inside goes first)
outer function — In a composition, the function applied second, to the inner function's output; in f(g(x)) the outer function is f. (first used: 7. Composition: the inside goes first)

Summary

A function gives each input exactly one output; two inputs may share an output.
A graph is a function when no vertical line meets it more than once.
To evaluate, put the input in parentheses everywhere the variable appears, then follow the order of operations.
For an expression as the input, substitute the whole expression and distribute to every term.
A polynomial takes every real number; a rational function drops the zeros of its denominator; a square root needs its radicand to be zero or more.
The range is the set of outputs: read it up and down on a graph, not left and right.
To add, subtract, multiply, or divide two functions, combine their outputs at the same input, and drop any input that makes a quotient's bottom zero.
A composition runs inside first, its order matters, and it is not a product.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

A school registrar's list pairs four student ID numbers with birth months: 1101 → March, 1102 → July, 1103 → March, 1104 → October. A second list pairs each month with the IDs born in it, so March → 1101 and 1103. Which list is a function, and why?
A ride service charges F(m) = 2m + 5 dollars for a trip of m miles. Find F(4). Then find F(a + 3) and check your formula with a = 1.
Find the domain of w(x) = (x − 2) ⁄ (x + 6) and of v(x) = √(x + 4). Write both in interval notation.
Let m(x) = 2x and n(x) = x − 3. Find (m ∘ n)(5) and (n ∘ m)(5). Are they equal?

Answers (worked reasoning — read after attempting)

The first list is a function. Each ID has exactly one month, even though 1101 and 1103 share March; sharing an output is allowed. The second list is not a function, because the one input March has two outputs, 1101 and 1103. That is a relation but not a function.
F(4) = 2 · 4 + 5 = 8 + 5 = 13 dollars. For F(a + 3), substitute the whole expression: 2(a + 3) + 5 = 2a + 6 + 5 = 2a + 11. Check with a = 1: the trip is 1 + 3 = 4 miles, which costs 13 dollars, and 2 · 1 + 11 = 13. They match. If you got 2a + 8, you forgot to multiply the 3 by 2.
For w, the denominator is zero when x + 6 = 0, so x = −6 is left out. The domain is (−∞, −6) ∪ (−6, ∞). For v, the radicand must be zero or more: x + 4 ≥ 0, so x ≥ −4. The domain is [−4, ∞), with a bracket because v(−4) = √0 = 0 is a real output.
(m ∘ n)(5): inside first, n(5) = 5 − 3 = 2, then m(2) = 2 · 2 = 4. (n ∘ m)(5): inside first, m(5) = 2 · 5 = 10, then n(10) = 10 − 3 = 7. They are not equal, 4 versus 7, because the order of a composition matters.

Next up in this module: the Readings & Resources page, the lecture tutorial, the practice set, and the quiz — and next week a straight line becomes a function whose slope rule tells you its rate of change.

Built with The Course Maker · samples.thecoursemaker.com