SAT — mathFoundational6 min read

Function Notation and Evaluation

Master function notation f(x) for the Digital SAT. Learn to evaluate functions, interpret f(a) = b, and work with function tables and graphs.

Tutor AI Team2026-02-25T01:10:50.877Z

Function notation — writing $f(x)$ instead of just $y$ — is the language of functions on the Digital SAT. Understanding what $f(x)$ means, how to evaluate it, and how to read function values from tables and graphs is essential for a wide range of SAT problems.

Function notation appears in algebra, data analysis, and even some geometry contexts. It's not a difficult concept, but the SAT tests it in many ways, so fluency is key.

Core Concepts

What Is Function Notation?

The notation $f(x)$ means "the function $f$ evaluated at $x$ ." It tells you the output when the input is $x$ .

$f(x) = 2x + 3$

means: take the input $x$ , multiply by 2, add 3 → that's the output.

$f(x)$ is NOT $f$ times $x$ . The parentheses indicate function evaluation, not multiplication.

Evaluating a Function

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

Example: If $f(x) = 3x^2 - 5x + 1$ , find $f(2)$ .

$f(2) = 3(2)^2 - 5(2) + 1 = 12 - 10 + 1 = 3$

Example: If $g(x) = \frac{x + 4}{x - 1}$ , find $g(3)$ .

$g(3) = \frac{3 + 4}{3 - 1} = \frac{7}{2}$

Evaluating with Expressions

You can input expressions, not just numbers.

Example: If $f(x) = x^2 + 1$ , find $f(a + 1)$ .

Replace $x$ with $(a + 1)$ :

$f(a + 1) = (a + 1)^2 + 1 = a^2 + 2a + 1 + 1 = a^2 + 2a + 2$

Reading Function Values from Tables

A table of values shows input-output pairs:

$x$	$f(x)$
0	5
1	8
2	11
3	14

From this table: $f(0) = 5$ , $f(2) = 11$ .

To find $x$ when $f(x) = 14$ : look for 14 in the $f(x)$ column → $x = 3$ .

Reading Function Values from Graphs

For a graphed function:

$f(a)$ = the y-coordinate when $x = a$ (go to $x = a$ , read up/down to the graph, read the y-value)
If $f(x) = b$ , find the x-value(s) where the graph has y-coordinate $b$

Notation Variations

$f(x)$ , $g(x)$ , $h(x)$ are different functions
$f(2)$ = output when input is 2
$f(x) = 5$ means "find the input(s) that produce output 5"
$f(0)$ = the y-intercept of the function

Function Equality

If $f(a) = g(a)$ , the two functions have the same output at $x = a$ . Graphically, this is where their graphs intersect.

Strategy Tips

Tip 1: Substitute Carefully

When evaluating, use parentheses around the substituted value: $f(-3) = 2(-3)^2$ , not $2 \cdot -3^2$ .

Tip 2: $f(0)$ Is the Y-Intercept

Remember: $f(0)$ gives you the value where the function crosses the y-axis.

Tip 3: Working Backwards

If the SAT gives you $f(a) = 10$ and asks for $a$ , set the function equal to 10 and solve for $a$ .

Tip 4: Multiple Functions

If $f(x) = x + 3$ and $g(x) = 2x$ , what is $f(g(4))$ ?

First: $g(4) = 2(4) = 8$

Then: $f(8) = 8 + 3 = 11$

Solution

Practice Problems

Problem 1

If $f(x) = -2x + 9$ , find $f(3)$ and $f(-1)$ .

Problem 2

If $g(x) = x^2 + 2x - 5$ , find $g(0)$ and $g(-3)$ .

Problem 3

If $h(x) = 5x - 2$ and $h(a) = 23$ , find $a$ .

Problem 4

Using the table: $f(1) = 4$ , $f(2) = 7$ , $f(3) = 10$ , $f(4) = 13$ . If $f(x) = 3x + b$ , find $b$ .

Problem 5

If $f(x) = x + 5$ and $g(x) = x^2$ , find $g(f(2))$ .

Problem 6

The function $f(x) = ax + 3$ passes through the point $(2, 11)$ . Find $a$ .

Want to check your answers and get step-by-step solutions?

Common Mistakes

Treating $f(x)$ as multiplication. $f(3)$ does NOT mean $f \times 3$ .
Not using parentheses when substituting negatives. $f(-2) = (-2)^2 = 4$ , not $-2^2 = -4$ .
Confusing $f(a)$ (find output) with $f(x) = a$ (find input). Read the question carefully.
Order of operations in composite functions. For $f(g(x))$ , evaluate $g(x)$ FIRST, then apply $f$ .
Misreading graph values. On SAT graphs, check the scale carefully before reading function values.

Frequently Asked Questions

Is $f(x)$ the same as $y$?

Yes, in most contexts. $y = f(x)$ . Both represent the output of the function.

What's the difference between $f(x)$ and $g(x)$?

They're just different function names, like using different variable names. Each can have a completely different rule.

What is $f(0)$?

It's the output when the input is 0 — which is the y-intercept of the function's graph.

Can $f(x)$ have more than one output for the same input?

No. By definition, a function gives exactly one output for each input. This is the vertical line test.

Do I need to know about domain and range for function notation questions?

Sometimes. The SAT may ask about values for which $f(x)$ is undefined (e.g., when a denominator is 0 or a value under a square root is negative).

Key Takeaways

✓
$f(x)$ means "plug $x$ into the function rule."
✓
To evaluate: replace every $x$ with the given value and compute.
✓
$f(a) = k$ means: set the function equal to $k$ and solve for $a$ .
✓
$f(0)$ is the y-intercept.
✓
For composite functions, evaluate inside-out: $f(g(x))$ → find $g(x)$ first, then apply $f$ .
✓
Read tables and graphs carefully — they're just another way of representing function values.

Tags:

#sat#math#algebra#functions#function-notation

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Core Concepts

What Is Function Notation?

Evaluating a Function

Evaluating with Expressions

Reading Function Values from Tables

Reading Function Values from Graphs

Notation Variations

Function Equality

Strategy Tips

Tip 1: Substitute Carefully

Tip 2: f(0)f(0)f(0) Is the Y-Intercept

Tip 3: Working Backwards

Tip 4: Multiple Functions

Tip 5: Don't Overthink

Worked Example: Example 1

Worked Example: Example 2

Worked Example: Example 3

Worked Example: SAT-Style

Worked Example: Example 5

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Common Mistakes

Frequently Asked Questions

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

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Tip 2: $f(0)$ Is the Y-Intercept