Lecture: Functions: Notation, Domain & Range
Lecture script · about 15–20 minutes spoken · Week 3
Opening
Picture a bowl shop on a busy street. A plain bowl costs 4 dollars, and each topping adds 3 dollars. You tell the cashier how many toppings you want, and the register shows one price. With 2 toppings it shows 10 dollars. With 5 toppings it shows 19 dollars. Nobody argues about it, because the rule never gives two prices for the same order. That little promise, one answer for each thing you put in, is the whole subject of this lecture.
Rules like that run a lot of daily life. A pharmacy chart turns a weight into one dose. A spreadsheet column turns one number into another. A ride fare turns miles into dollars. In each case it matters which inputs the rule can accept, because a nurse, a shop owner, or a programmer who feeds a rule a value it cannot handle gets nonsense back.
In Week 1, you learned to evaluate an expression with the order of operations, and to be careful with a negative base. In Week 2, you learned to solve for an unknown and to write a set of numbers in interval notation. This week we give rules a name, we plug numbers and expressions into them, and we ask which inputs they accept and which outputs they produce. Next week, in Week 4, the simplest rule of all, the straight line, gets its own lecture. I'll walk through seven ideas, and I'll stop after each one so you can check yourself.
1. What a function is: one input, one output
Let's go back to the bowl shop. You say a number of toppings, and the register says a price. You'd be annoyed if 2 toppings rang up as 10 dollars one minute and 13 dollars the next. The shop's rule makes a promise: one price for each order. In algebra, a rule that keeps that promise has a name. 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. The thing you feed in is the input, and the thing you get back is the output.
A function can show up in different clothes. It can be a list of arrows, from each topping count to its price: 0 goes to 4, 1 goes to 7, 2 goes to 10, and so on up to 5, which goes to 19. It can be a table. Picture a 100-gram sample in a lab that loses half its mass each period. After one period there are 50 grams, then 25, then 12.5, then 6.25. Each period count sits beside one mass. And it can be a graph.
Here's the part that surprises people. Two inputs are allowed to land on the same output. Take the squaring rule. Negative 2 squared is 4, and 2 squared is also 4. That's fine. Think of birthdays. Two people can share a birthday, but one person can't have two birthdays. Sharing an output is allowed; splitting an input is not.
For a graph, there's a quick check. A 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. Picture a U-shaped curve that opens upward, the graph of y equals x squared. Slide a vertical line across it from left to right. Everywhere you stop, the line touches the curve once. Now turn the U on its side so it opens to the right. That's the graph of x equals y squared. Stop your vertical line at x equals 4. It touches the curve twice, once at height 2 and once at height negative 2.
Pause and try this. On that sideways curve, the input 4 pairs with both 2 and negative 2. Is that sideways curve a function?
Hold that picture of the sideways U for a moment.
Here's the answer. No, it isn't, because the one input 4 has two outputs, 2 and negative 2. Squaring either one gives 4, so both points sit on the curve. That makes it a relation, a set of input-output pairs, but not a function.
2. Function notation: one name, one output
Once a rule has a name, talking about it gets much faster. Instead of saying "the output of the rule when the input is negative 2," you write the letter f, then the input in parentheses. You read it out loud as "f of negative 2." Function notation is the way of writing a function's output as f of x, where f names the function and x is the input; f of x is one output value, not f multiplied by x.
This week's main function takes an input, squares it, multiplies by 3, subtracts 2 times the input, and adds 5. In symbols, that's 3 x squared, minus 2 x, plus 5. Let's evaluate it at negative 2, slowly. The move that saves you is this: wrap the input in parentheses before you do anything else.
First, square negative 2. Negative 2 times negative 2 is positive 4. Next, multiply by 3, which gives 12. Then the middle term: negative 2 times negative 2 is positive 4 again. Finally, add the 5. So 12 plus 4 plus 5 is 21. That means f of negative 2 is 21.
Two more inputs, quickly. At 0, the first two terms vanish and you're left with 5, so f of 0 is 5. At 3, you square 3 to get 9, times 3 is 27, minus 6 is 21, plus 5 is 26. So f of 3 is 26. Across the six whole-number inputs from negative 2 to 3, the outputs are 21, 10, 5, 6, 13, and 26.
Here's a classic wrong turn, and I want you to see it coming. A lot of students let the minus sign slip outside the square. They write 3 times negative 4, get negative 12, and finish with negative 3. The parentheses say square the whole number, minus sign and all.
Pause and try this. Someone tells you f of negative 2 is negative 3. Which step went wrong, and what should it have been?
Think about the very first step, the square.
Here's the answer. They squared the 2 and kept the minus sign outside, getting negative 4. Negative 2 times negative 2 is positive 4, so the first term is positive 12, not negative 12, and f of negative 2 is 21.
3. Algebraic inputs: substitute the whole expression
Back to the bowl shop. Suppose a friend always orders 2 more toppings than you, and you call your topping count a. Your friend's price is the shop's rule with a plus 2 fed in. Let's call the shop's rule g. It multiplies the input by 3 and adds 4. An algebraic input is an input that is an expression with a letter in it, such as a plus 2 or negative x, rather than a single number; you substitute the whole expression wherever the variable appears.
So g of a plus 2 means 3 times the whole quantity a plus 2, plus 4. Distribute the 3 to both pieces: 3 times a is 3a, and 3 times 2 is 6. Then 6 plus 4 is 10. So g of a plus 2 is 3a plus 10.
Here's how I'd check it. If you order 1 topping, your friend orders 3. Three toppings cost 3 times 3 plus 4, which is 13 dollars. The formula gives 3 times 1 plus 10, which is also 13. They agree, and an agreement like that is how you know you're right.
The same move works for any input you can write down. Feed in negative x, and g gives negative 3x plus 4. Feed in x plus h, where h is some small amount added to the input, and you get 3x plus 3h plus 4.
Pause and try this. A classmate writes g of a plus 2 as 3a plus 6. Using a equal to 1, show that something is off.
Put 1 in for a and compare the two prices.
Here's the answer. Their formula gives 3 plus 6, which is 9 dollars, but 3 toppings really cost 13 dollars. They multiplied only the a by 3 and forgot the 2; the 3 has to multiply the whole input.
4. Domain: which inputs does the rule accept?
A calculator gives up on two kinds of requests. It won't divide by zero, and it won't take the square root of a negative number. Any input that forces one of those is an input the rule can't 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. First, a polynomial, a rule built from numbers and whole-number powers of x like our function f, accepts every real number. Second, 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. Take the rule that divides x plus 1 by x minus 3. The bottom is zero when x is 3, so 3 is out. Every other number is in. Third, a square root needs the expression under it, the radicand, to be zero or more. Take the square root of x minus 5. You need x minus 5 to be at least 0, so x has to be at least 5. The domain starts at 5, includes 5, and runs forever to the right.
A real setting can shrink a domain further. The formula for the bowl price accepts any real number, but the shop only sells whole toppings from 0 to 5. That smaller set is the practical domain: the inputs that make sense in a real setting, which can be fewer than the formula allows.
Pause and try this. For the square root of x minus 5, is 5 itself in the domain?
Try putting 5 into the rule and see what comes out.
Here's the answer. Yes, 5 is in. Five minus 5 is 0, and the square root of 0 is 0, a real number. A zero under a square root is fine; a zero in a denominator is not. So the domain begins with a square bracket at 5.
5. Range: what comes out?
The domain asks what can go in. The range asks what actually comes out. The range is the set of every output a function actually produces as its input runs through the domain. For the bowl shop, the prices that come out are 4, 7, 10, 13, 16, and 19 dollars, and nothing else.
On a graph, the domain runs left and right, and the range runs up and down. So to read a range, look at how low and how high the graph goes. The U-shaped squaring graph bottoms out at 0 and climbs forever, because a square is never negative. Its range starts at 0 and goes up without end.
Now a story with some motion in it. An object is launched straight up from a platform 80 feet high. At launch it's at 80 feet. After 1 second it's at 128 feet, and after 2 seconds it reaches 144 feet. Then it falls back: 128 feet at 3 seconds, 80 feet at 4, and 0 at 5 seconds, when it lands. The flight lasts from 0 to 5 seconds, so that's the practical domain. One caveat: this model ignores air resistance. It's the standard simplified model, good for learning, not a promise about a windy day.
Pause and try this. During that flight, what is the lowest height and the highest height the object reaches?
Picture the path rising, peaking, and falling to the ground.
Here's the answer. The lowest height is 0, when it lands, and the highest is 144 feet, at 2 seconds. So the range runs from 0 to 144, both included. A common slip is to answer with 0 to 5, but those are the times, which belong to the domain, not the heights.
6. Combine two functions: add, subtract, multiply, divide
Suppose two rules each give a number for the same input. You can add those two numbers, subtract them, multiply them, or divide them. 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.
Let's use our two named rules, f and g, at the input 2. Earlier we found that f of 2 is 13, and the bowl rule gives g of 2 equal to 10. Their sum at 2 is 13 plus 10, which is 23. Their difference is 13 minus 10, which is 3. At the input 1, f gives 6 and g gives 7, so their product there is 42. And the quotient of f by g at 2 is 13 divided by 10, which is 1.3.
The quotient carries a warning label. You can't divide by an output of zero, so any input that makes g equal 0 gets left out. That happens at negative 4 thirds.
The slip I see most here is with subtraction. When you subtract g from f, the minus sign has to reach every piece of g, not only the first one.
Pause and try this. Someone subtracts g from f, writes a formula, and it gives 11 at the input 2. Should you trust it?
Compare it with the outputs you already know at 2.
Here's the answer. No. At 2, f is 13 and g is 10, so the difference has to be 13 minus 10, which is 3. A formula that gives 11 dropped the minus sign on the plus 4 inside g.
7. Composition: the inside goes first
Go back to the register one more time. The shop gives students 2 dollars off. To find your cost, you first find the price, and then you take off the discount. One rule's output becomes the next rule's input. A composition is using one function's output as another function's input; the composition of f and g, written f of g of x, applies g first and then applies f to the result. The rule applied first is the inner function, and the rule applied second is the outer function.
With 3 toppings, the price is 13 dollars, and the discount brings it to 11. Now flip the order. Take 2 off the topping count first, so 3 becomes 1, and then price it. That gives 7 dollars, which is a different deal. Order matters.
Here's a small pair to see it cleanly. One rule squares its input. The other adds 1. Start at 2 and add 1 first: you get 3, and 3 squared is 9. Now square first: 2 squared is 4, and 4 plus 1 is 5. Same two rules, same start, and the answers are 9 and 5. And if you multiply the two outputs at 2, 4 times 3, you get 12, a third number. A composition is not a product.
With our named rules, it's even more dramatic. Put negative 2 into g first: 3 times negative 2 plus 4 is negative 2. Then f of negative 2 is 21. Go the other way: f of negative 2 is 21, and g of 21 is 3 times 21 plus 4, which is 67.
Pause and try this. For f of g of negative 2, which rule do you apply first, and why?
Read the expression from the inside out.
Here's the answer. You apply g first, because g sits inside the parentheses. Its output, negative 2, then goes into f, which gives 21. Doing f first gives 67 instead, a very different number.
Closing
Let me pull it together. A function gives each input exactly one output, and a vertical line can check a graph for you. Function notation names the output, and a negative input always goes in parentheses. An algebraic input is substituted whole, and the multiplier reaches every term. The domain is what the rule accepts: every real number for a polynomial, no zero denominators, and nothing negative under a square root. The range is what comes out, read up and down on a graph. You can add, subtract, multiply, and divide functions output by output. And a composition runs from the inside out.
Next in this module come the slide deck, the Readings & Resources page, the lecture tutorial, the practice set, and the quiz. Next week, in Week 4, Linear Functions & Their Graphs, the straight line becomes a function, and you'll read its slope and its starting value right off the rule.
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