Before you read

Answer first — the lesson below explains it.

A shipping company's cost function is $f (x) = 4 x - 3$ , where $x$ is the weight in pounds. What is the cost to ship a $5$ -pound package? (Find $f (5)$ .)

A
$17$
B
$23$
C
$2$
D
$20$

Check your answer

Answer:

Before you read

Answer first — the lesson below explains it.

A profit model is $g (x) = - 2 x + 10$ , where $x$ is months since launch. What was the profit $3$ months before launch? (Find $g (- 3)$ .)

A
$4$
B
$16$
C
$- 16$
D
$7$

Check your answer

Answer:

Theory

What Is a Function?

A function is a rule that takes an input and produces exactly one output.

Think of it as a machine:
- You feed in a number (input $x$ )
- The machine applies a rule
- Out comes exactly one result (output $y$ )

We write $f(x)$ (read "f of x") to mean "the output of function $f$ when the input is $x$ ."

f(x) = 2x + 1: f(3) = 7

Important: $f(x)$ does NOT mean $f$ times $x$ . The parentheses here mean "function of," not multiplication.

Quick check

You just read it. Can you apply it?

A pricing function is $f (x) = 3 x + 1$ . If a customer orders $2 a$ units, what is the total price $f (2 a)$ ?

A
$6 a + 1$
B
$6 a + 2$
C
$3 a + 2$
D
$5 a$

Check your answer

Answer:

$f (2 a) = 3 (2 a) + 1 = 6 a + 1$

Theory

Evaluating a Function

To evaluate $f(a)$ , replace every $x$ in the formula with $a$ .

This is the single most important skill for function questions on the SAT.

Example 1

If $f(x) = 3x + 7$ , find $f(4)$ .

Replace $x$ with $4$ :

$f(4) = 3(4) + 7 = 12 + 7 = 19$

f(4) = 19

Example 2

If $g(x) = x^2 - 5x + 2$ , find $g(-1)$ .

Replace $x$ with $-1$ :

$g(-1) = (-1)^2 - 5(-1) + 2 = 1 + 5 + 2 = 8$

g(-1) = 8

Example 3

If $f(x) = 2x + 1$ , find $f(a + 3)$ .

Replace $x$ with $(a + 3)$ :

$f(a + 3) = 2(a + 3) + 1 = 2a + 6 + 1 = 2a + 7$

f(a + 3) = 2a + 7

Quick check

You just read it. Can you apply it?

A commission function is $h (x) = 7 x - 4$ . If Aiden's commission is $$31$ , how many sales $k$ did Aiden make? ( $h (k) = 31$ )

A
$3$
B
$5$
C
$\frac{27}{7}$
D
$7$

Check your answer

Answer:

$7 k - 4 = 31$ $\to$ $7 k = 35$ $\to$ $k = 5$ .
Check: $h (5) = 7 (5) - 4 = 31$ ✓

Theory

Working Backwards: Finding the Input

Sometimes the SAT gives you the output and asks for the input:

"If $f(x) = 4x - 3$ and $f(a) = 17$ , find $a$ ."

This just means: set the formula equal to $17$ and solve:
$4a - 3 = 17 \quad \Rightarrow \quad 4a = 20 \quad \Rightarrow \quad a = 5$

Example 1

If $f(x) = 5x + 2$ and $f(k) = 27$ , what is $k$ ?

Set $f(k) = 27$ :

$5k + 2 = 27$

$5k = 25$

$k = 5$

k = 5

Quick check

You just read it. Can you apply it?

The function $f$ is defined by the table below.

$x$	$f (x)$
$- 1$	$4$
$0$	$7$
$1$	$10$
$2$	$13$

What is $f (1) + f (0)$ ?

A
$1$
B
$10$
C
$17$
D
$11$

Check your answer

Answer:

From the table: $f (1) = 10$ and $f (0) = 7$ .
$f (1) + f (0) = 10 + 7 = 17$ .

Theory

Reading Function Values from Tables and Graphs

On the SAT, functions aren't always given as formulas. You may see:

Tables: Find the row where the input matches, read the output.

$x$	$f(x)$
1	5
2	8
3	11

$f(2) = 8$ (look at the row where $x = 2$ )

Graphs: Find the input on the x-axis, go up/down to the curve, read the y-value.

$f(3) =$ the y-coordinate of the point on the graph where $x = 3$ .

Quick check

You just read it. Can you apply it?

If $f (x) = 2 x + 5$ , which expression is equal to $f (x + 3) - f (x)$ ?

A
$3$
B
$6$
C
$2 x + 3$
D
$8$

Check your answer

Answer:

$f (x + 3) = 2 (x + 3) + 5 = 2 x + 6 + 5 = 2 x + 11$
$f (x) = 2 x + 5$
$f (x + 3) - f (x) = (2 x + 11) - (2 x + 5) = 6$

Common Mistakes

Common Mistakes to Avoid

Treating

f(x)

as multiplication

If $f(x) = x + 3$ , writing $f(2) = f \times 2 = 2f$

f(2)

means substitute

2

for

x

f(2) = 2 + 3 = 5

Forgetting parentheses when substituting negative numbers

$f(x) = x^2$ , then $f(-3) = -3^2 = -9$ $\leftarrow$ WRONG!

f(-3) = (-3)^2 = 9

. The parentheses matter for exponents!

Evaluating

f(a+b)

f(a) + f(b)

If $f(x) = x^2$ , then $f(1+2) = f(3) = 9$ , but $f(1) + f(2) = 1 + 4 = 5$

In general,

f(a+b) \neq f(a) + f(b)

. Always substitute the entire expression.

Quick check

You just read it. Can you apply it?

If $f (x) = a x + b$ where $a$ and $b$ are constants, $f (2) = 11$ , and $f (5) = 23$ , what is $f (0)$ ?

A
$3$
B
$4$
C
$7$
D
$11$

Check your answer

Answer:

$f (2) = 2 a + b = 11$
$f (5) = 5 a + b = 23$
Subtract: $3 a = 12$ $\to$ $a = 4$ .
Then $b = 11 - 2 (4) = 3$ .
$f (0) = a (0) + b = b = 3$ .

Strategy

SAT Strategy Tip

Function evaluation questions are easy points on the SAT — they just test whether you can substitute carefully. The SAT makes them tricky by using expressions like $f(2a)$ , $f(x+1)$ , or $f(-x)$ as inputs. Just replace $x$ with the entire expression, use parentheses, and simplify.

After reading

Same question as before. What do you say now?

A shipping company's cost function is $f (x) = 4 x - 3$ , where $x$ is the weight in pounds. What is the cost to ship a $5$ -pound package? (Find $f (5)$ .)

A
$17$
B
$23$
C
$2$
D
$20$

Check your answer

Answer:

$f (5) = 4 (5) - 3 = 20 - 3 = 17$

After reading

Same question as before. What do you say now?

A profit model is $g (x) = - 2 x + 10$ , where $x$ is months since launch. What was the profit $3$ months before launch? (Find $g (- 3)$ .)

A
$4$
B
$16$
C
$- 16$
D
$7$

Check your answer

Answer:

$g (- 3) = - 2 (- 3) + 10 = 6 + 10 = 16$

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