SAT function notation, composition and shifts

Q: What does f(x) mean?

It is the output of the function f when the input is x. f(4) = 9 means "put in 4, get out 9", and on a graph it is the point (4, 9). The parentheses do not mean multiplication here.

Q: Is f(g(x)) the same as g(f(x))?

Usually not. In f(g(x)) you apply g first, then f. Switching the order changes the result for most pairs of functions, so always work from the inside parentheses out.

The key idea

Function notation is a machine label. $f(x) = 2x^2 - 3x + 1$ says what f does to any input. Whatever is inside the parentheses replaces every x, even if it is an expression:

f(a + 1) = 2(a + 1)^2 - 3(a + 1) + 1

A composition $f(g(x))$ works inside out: find $g(x)$ first, then feed that output into f.

How the graph of y = f(x) moves
New function	What happens to the graph	Point (a, b) moves to
$f(x) + k$	Up k units	$(a, b + k)$
$f(x - h)$	Right h units (left if h is negative)	$(a + h, b)$
$-f(x)$	Flip over the x-axis	$(a, -b)$
$f(-x)$	Flip over the y-axis	$(-a, b)$
$c \cdot f(x)$	Vertical stretch by c	$(a, cb)$

Changes outside the parentheses move the graph up and down the way you expect. Changes inside the parentheses move it sideways the opposite way: $f(x - 2)$ goes right, not left.

Worked examples

Example 1: evaluate

Problem If $f(x) = 2x^2 - 3x + 1$ , what is $f(-2)$ ?

Replace every x with $-2$ , in parentheses.
$2(-2)^2 - 3(-2) + 1$
Square first: $(-2)^2 = 4$ . Then multiply.
$8 + 6 + 1 = 15$

Answer $f(-2) = 15$

Example 2: composition

Problem Let $f(x) = 3x - 1$ and $g(x) = x^2 + 2$ . Find $f(g(2))$ and $g(f(2))$ .

Inside first: $g(2) = 2^2 + 2 = 6$ . Then $f(6)$ :
$3(6) - 1 = 17$
Other order: $f(2) = 3(2) - 1 = 5$ . Then $g(5)$ :
$5^2 + 2 = 27$
The order matters. $f(g(2))$ and $g(f(2))$ are usually different.

Answer $f(g(2)) = 17$ and $g(f(2)) = 27$

Example 3: a shifted graph

Problem The graph of $y = f(x)$ passes through $(4, -1)$ . Which point must be on the graph of $y = f(x - 3) + 2$ ?

Inside the parentheses, $x - 3$ moves the graph 3 units right.
$4 + 3 = 7$
Outside, $+2$ moves it 2 units up.
$-1 + 2 = 1$
Check: at $x = 7$ , the new function is $f(7 - 3) + 2 = f(4) + 2 = -1 + 2 = 1$ .

Answer $(7, 1)$

Example 4 (SAT-hard): a function given in disguise

Problem If $f(x + 1) = 2x + 7$ for all x, what is $f(5)$ ?

You need the input $x + 1$ to equal 5.
$x + 1 = 5$
So use $x = 4$ on the right side.
$2(4) + 7 = 15$
Another way: let $u = x + 1$ , so $x = u - 1$ and $f(u) = 2(u - 1) + 7 = 2u + 5$ . Then $f(5) = 15$ . Plugging 5 straight into $2x + 7$ gives the trap answer 17.

Answer $f(5) = 15$

Common mistakes

Squaring a negative without parentheses. $-2^2$ is $-4$ , but $(-2)^2$ is 4. Fix: always put the input in parentheses when you substitute.
Working a composition outside in. $f(g(2))$ starts with g. Fix: find the innermost output first.
Moving the graph the wrong way. $f(x - 3)$ moves right, not left. Fix: ask what x makes the inside equal the old input. Here $x - 3 = 4$ when $x = 7$ .
Reading f(x) as f times x. $f(3)$ is an output, not $3f$ . Fix: say "f of 3" out loud.
Plugging the given number into the wrong place. For $f(x + 1) = 2x + 7$ , x is not the input. Fix: solve for the x that makes the whole input match.

Quick methods

Practice

5 SAT-style questions

If $g(x) = x^2 - 4x$ , what is $g(-3)$ ?
1. $-21$
2. $-3$
3. $3$
4. $21$
Show answer
Answer: $21$
$(-3)^2 - 4(-3) = 9 + 12 = 21$ . $-3$ comes from $9 - 12$ , which forgets that $-4 \times -3$ is positive.
If $f(x) = 4x + 1$ and $g(x) = 2x - 3$ , which expression is $f(g(x))$ ?
1. $8x - 11$
2. $8x - 1$
3. $8x^2 - 10x - 3$
4. $6x - 2$
Show answer
Answer: $8x - 11$
Replace x in f with $2x - 3$ : $4(2x - 3) + 1 = 8x - 12 + 1 = 8x - 11$ . $8x - 1$ is $g(f(x))$ , the other order. $8x^2 - 10x - 3$ multiplies the functions, and $6x - 2$ adds them.
The graph of $y = f(x)$ contains the point $(2, 5)$ . Which point must be on the graph of $y = f(x + 4) - 1$ ?
1. $(6, 4)$
2. $(-2, 4)$
3. $(-2, 6)$
4. $(6, 6)$
Show answer
Answer: $(-2, 4)$
$x + 4$ inside moves the graph 4 units left, so x goes from 2 to $-2$ . The $-1$ outside moves it down 1. Check: $f(-2 + 4) - 1 = f(2) - 1 = 4$ .
The function h is defined by $h(x) = kx^3 - 2$ , where k is a constant. If $h(2) = 30$ , what is k?
1. $2$
2. $3.75$
3. $4$
4. $16$
Show answer
Answer: $4$
$k(2)^3 - 2 = 30$ , so $8k = 32$ and $k = 4$ . 3.75 forgets to add the 2 back before dividing.
Student-produced response: the function f is defined by $f(x) = \frac{x + 3}{x - 1}$ . If $f(a) = 3$ , what is a?
Show answer
Answer: 3
Set $\frac{a + 3}{a - 1} = 3$ and multiply both sides by $a - 1$ : $a + 3 = 3a - 3$ , so $6 = 2a$ and $a = 3$ . Check: $f(3) = \frac{6}{2} = 3$ .

Frequently asked questions

What does f(x) mean?

It is the output of the function f when the input is x. $f(4) = 9$ means "put in 4, get out 9", and on a graph it is the point $(4, 9)$ . The parentheses do not mean multiplication here.

Why does f(x - 2) move the graph to the right?

The new function reaches each old output 2 units later. To get the old value $f(0)$ , you now need $x - 2 = 0$ , so $x = 2$ . Every point slides 2 units to the right.

How do I read a function from a table on the SAT?

Each row is one input and its output, so each row is a point on the graph. To find $f(3)$ , look up the row with input 3. To solve $f(x) = 10$ , look for the output 10 and read its input.

Is f(g(x)) the same as g(f(x))?

Usually not. In $f(g(x))$ you apply g first, then f. Switching the order changes the result for most pairs of functions, so always work from the inside parentheses out.

Sources

College Board: SAT Math, Advanced Math skills (nonlinear functions), accessed October 1, 2026

Functions and function notation on the SAT

The key idea

Worked examples

Common mistakes

Quick methods

Practice

Frequently asked questions

Sources

Try asking Ducky

The key idea

Worked examples

Common mistakes

Quick methods

Practice

Frequently asked questions

Sources

Related

Try asking Ducky