What this lesson covers
- f(x) is a machine: whatever sits in the slot gets substituted everywhere x appears
- Composites run from the innermost parentheses outward
- In a table, read down for f(4) and read back up when they give you the output
- f(x + 1) puts x + 1 into the machine; f(x) + 1 adds after the machine is done
Chapters
- 0:00f(3) is not f times 3
- 0:26The machine: input, rule, output
- 1:00Composing: f(g(2)) inside out
- 1:45Reading a function table
- 2:21The parentheses trap: f(x + 1)
- 2:59Your turn
- 3:39Three traps
- 4:05Recap
Lesson transcript
The narration of the video, word for word, under its chapter headings.
0:00f(3) is not f times 3
Welcome to SAT Climb. The single most expensive misread in SAT algebra is this one. f of 3 is not f times 3. There is no multiplication anywhere in that symbol. It is an instruction. Put 3 into the machine and see what comes out. Get that one idea right and function notation stops being hard.
0:26The machine: input, rule, output
Here is the machine. f of x equals two x plus one. The x in the parentheses is the input slot. Whatever sits in that slot gets substituted everywhere x appears in the rule. So f of 3 is two times 3 plus one, which is 7. f of 10 is twenty one. f of a is two a plus one. The name of the input does not matter. The slot does. Read f of x equals two x plus one as, whatever you hand me, I double it and add one.
1:00Composing: f(g(2)) inside out
Now two machines. f of x is two x plus one. g of x is x squared minus three. Find f of g of 2. Start at the innermost parentheses, always. g of 2 is 2 squared minus 3, which is 1. That 1 is now the input to f. f of 1 is two times one plus one, which is 3. The answer is 3. Watch what happens if you go the other way. g of f of 2. f of 2 is 5, and g of 5 is twenty five minus three, which is twenty two. Same two functions, opposite order, 3 versus twenty two. Inside out is not a preference. It is the definition.
1:45Reading a function table
Tables work the same way. Here is a table of x and f of x. To find f of 4, find 4 in the input row and read straight down. f of 4 is 2. Now the reverse. If the question says f of x equals 8, they gave you the output. Find 8 in the bottom row and read back up. x is 3. And a composite in a table is just two lookups. f of f of 4. Inside first, f of 4 is 2, so this is f of 2, which is 3. No formula required. Only careful reading.
2:21The parentheses trap: f(x + 1)
One more trap the test loves. f of x plus one, with the plus one inside the parentheses, is not the same as f of x, plus one. Take f of x equals two x plus one. f of the quantity x plus one means the whole thing x plus one goes into the machine. Two times x plus one, plus one, is two x plus three. But f of x, plus one, means run the machine first and add one after. That is two x plus two. Different expressions. Inside the parentheses means into the machine. Outside means after the machine is finished.
2:59Your turn
Your turn. f of x is x squared minus two. g of x is three x. Find g of f of 3. Pause and work it. Inside first. f of 3 is nine minus two, which is 7. Now that 7 goes into g. g of 7 is three times seven, twenty one. The answer is twenty one. If you got sixty three, you ran it outside in, three times three is nine, nine squared minus two. That is f of g of 3, a different question.
3:39Three traps
Three traps. One, treating f of x as multiplication and distributing the f. There is no product there at all. Two, composing outside in. The innermost parentheses always run first. Three, sliding the plus one out of the parentheses. Inside the parentheses is the input. Outside is what you do to the output.
4:05Recap
Function notation, solved. f of x is a machine, not multiplication. Substitute into the slot, and when functions are nested, work from the inside out. Start free at satclimb.com.
Read the written version: the Nonlinear functions strategy guide, then try 25 hard Nonlinear functions questions with full explanations. Both are free, no account needed.