# Functions and function notation on the SAT

Canonical: https://duckyhelper.com/learn/sat-math/functions/
Updated: 2026-10-01

A function takes an input and gives exactly one output. \(f(3)\) means the output when the input is 3: replace every x with 3. The SAT asks you to evaluate functions, read them from graphs and tables, combine them like \(f(g(x))\), and shift graphs: \(f(x - 2)\) moves the graph 2 units right. Read \(f(a) = b\) as "the point \((a, b)\) is on the graph".

## 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 \(f(-2)\)?

1. Replace every x with \(-2\), in parentheses.

   $$
   2(-2)^2 - 3(-2) + 1
   $$
2. 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 \(g(f(2))\).

1. Inside first: \(g(2) = 2^2 + 2 = 6\). Then \(f(6)\):

   $$
   3(6) - 1 = 17
   $$
2. Other order: \(f(2) = 3(2) - 1 = 5\). Then \(g(5)\):

   $$
   5^2 + 2 = 27
   $$
3. 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 \(y = f(x - 3) + 2\)?

1. Inside the parentheses, \(x - 3\) moves the graph 3 units right.

   $$
   4 + 3 = 7
   $$
2. Outside, \(+2\) moves it 2 units up.

   $$
   -1 + 2 = 1
   $$
3. 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 \(f(5)\)?

1. You need the input \(x + 1\) to equal 5.

   $$
   x + 1 = 5
   $$
2. So use \(x = 4\) on the right side.

   $$
   2(4) + 7 = 15
   $$
3. 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

> **Tip: Follow one point**
>
> For shift questions, pick one point on the original graph and move it. It is faster and safer than redrawing the whole graph, and it is exactly how the transformation works.

> **Note: Desmos for function values**
>
> Define \(f(x) = 2x^2 - 3x + 1\) on one line of Desmos, then type \(f(-2)\) on the next line to see 15. You can also type \(f(g(2))\) once both are defined. It is a good check when the arithmetic is messy.

## Practice

**5 SAT-style questions**

1. If \(g(x) = x^2 - 4x\), what is \(g(-3)\)?
   A. \(-21\)
   B. \(-3\)
   C. \(3\)
   D. \(21\)

   Answer: \(21\). \((-3)^2 - 4(-3) = 9 + 12 = 21\). \(-3\) comes from \(9 - 12\), which forgets that \(-4 \times -3\) is positive.

2. If \(f(x) = 4x + 1\) and \(g(x) = 2x - 3\), which expression is \(f(g(x))\)?
   A. \(8x - 11\)
   B. \(8x - 1\)
   C. \(8x^2 - 10x - 3\)
   D. \(6x - 2\)

   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.

3. The graph of \(y = f(x)\) contains the point \((2, 5)\). Which point must be on the graph of \(y = f(x + 4) - 1\)?
   A. \((6, 4)\)
   B. \((-2, 4)\)
   C. \((-2, 6)\)
   D. \((6, 6)\)

   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: \(f(-2 + 4) - 1 = f(2) - 1 = 4\).

4. The function h is defined by \(h(x) = kx^3 - 2\), where k is a constant. If \(h(2) = 30\), what is k?
   A. \(2\)
   B. \(3.75\)
   C. \(4\)
   D. \(16\)

   Answer: \(4\). \(k(2)^3 - 2 = 30\), so \(8k = 32\) and \(k = 4\). 3.75 forgets to add the 2 back before dividing.

5. Student-produced response: the function f is defined by \(f(x) = \frac{x + 3}{x - 1}\). If \(f(a) = 3\), what is a?

   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)](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/advanced), accessed 2026-10-01

## Related

- [Quadratics on the SAT](https://duckyhelper.com/learn/sat-math/quadratics/)
- [Linear functions and slope on the SAT](https://duckyhelper.com/learn/sat-math/linear-functions/)
- [Transformations of functions](https://duckyhelper.com/learn/algebra-2/function-transformations/)
- [Functions, function notation, domain and range](https://duckyhelper.com/learn/algebra-1/functions/)
- [SAT Math study guides](https://duckyhelper.com/learn/sat-math/)

## Try asking Ducky

- "Why does the graph move right when there's a minus sign?"
- "Can you check my composition? I think I did it in the wrong order."
- "Give me a table question and let me find f(3)."

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