Functions

Master function notation, evaluation, and transformations for SAT

Written and reviewed by the Study Mondo Education TeamLast updated February 16, 2026

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Functions (SAT)

Function Notation

$f(x) = 2x + 3$

Reading: "f of x equals 2x plus 3"

Evaluating: $f(5) = 2(5) + 3 = 13$

Types of SAT Function Questions

1. Evaluation

"If $f(x) = x^2 - 3$ , what is $f(4)$ ?"

Answer: $f(4) = 16 - 3 = 13$

2. Composition

$(f \circ g)(x) = f(g(x))$

If $f(x) = 2x$ and $g(x) = x + 1$ : $f(g(3)) = f(4) = 8$

3. Finding Input

"If $f(x) = 3x - 1$ and $f(a) = 14$ , what is $a$ ?"

Solve: $3a - 1 = 14 \Rightarrow a = 5$

Domain and Range

Domain: All possible input values (x-values)
Range: All possible output values (y-values)

SAT Function Tricks

Watch for:

$f(x + 2)$ vs $f(x) + 2$ (very different!)
Questions asking for $2f(3)$ $2 f (3)$ when they give you $f(3) = 5$ $f (3) = 5$
- Answer: $2(5) = 10$ , not $f(6)$ !

📚 Practice Problems

1Problem 1easy

❓ Question:

If $f(x) = 3x - 4$ , what is $f(6)$ ?

💡 Show Solution

Solution:

Substitute $x = 6$ : $f(6) = 3(6) - 4$ $f(6) = 18 - 4$ $f(6) = 14$

Answer: $14$

2Problem 2easy

❓ Question:

If $f(x) = 3x - 4$ , what is $f(6)$ ?

💡 Show Solution

Solution:

Substitute $x = 6$ : $f(6) = 3(6) - 4$ $f(6) = 18 - 4$ $f(6) = 14$

Answer: $14$

3Problem 3medium

❓ Question:

If $g(x) = x^2 + 2x$ , what is $g(-3)$ ?

💡 Show Solution

Solution:

Substitute $x = -3$ : $g(-3) = (-3)^2 + 2(-3)$ $g(-3) = 9 - 6$ $g(-3) = 3$

Answer: $3$

SAT Tip: Be careful with negatives! $(-3)^2 = 9$

4Problem 4medium

❓ Question:

If $g(x) = x^2 + 2x$ , what is $g(-3)$ ?

💡 Show Solution

Solution:

Substitute $x = -3$ : $g(-3) = (-3)^2 + 2(-3)$ $g(-3) = 9 - 6$ $g(-3) = 3$

Answer: $3$

SAT Tip: Be careful with negatives! $(-3)^2 = 9$

5Problem 5hard

❓ Question:

If $h(x) = 2x + 5$ and $h(a) = 17$ , what is the value of $a$ ?

💡 Show Solution

Solution:

Set up the equation: $h(a) = 17$ $2a + 5 = 17$

Solve: $2a = 12$ $a = 6$

Answer: $a = 6$

Check: $h(6) = 2(6) + 5 = 17$ ✓

6Problem 6hard

❓ Question:

If $h(x) = 2x + 5$ and $h(a) = 17$ , what is the value of $a$ ?

💡 Show Solution

Solution:

Set up the equation: $h(a) = 17$ $2a + 5 = 17$

Solve: $2a = 12$ $a = 6$

Answer: $a = 6$

Check: $h(6) = 2(6) + 5 = 17$ ✓

7Problem 7easy

❓ Question:

If $f(x) = 3x - 7$ , what is the value of $f(5)$ ?

💡 Show Solution

Step 1: Replace every $x$ with $5$ in the function: $f(5) = 3(5) - 7$

Step 2: Evaluate: $f(5) = 15 - 7 = 8$

Answer: $f(5) = 8$

Key concept: $f(5)$ means "plug in 5 for $x$ ." Function notation is just a way to name the output.

8Problem 8easy

❓ Question:

If $f(x) = 3x - 7$ , what is the value of $f(5)$ ?

💡 Show Solution

Step 1: Replace every $x$ with $5$ in the function: $f(5) = 3(5) - 7$

Step 2: Evaluate: $f(5) = 15 - 7 = 8$

Answer: $f(5) = 8$

Key concept: $f(5)$ means "plug in 5 for $x$ ." Function notation is just a way to name the output.

9Problem 9medium

❓ Question:

If $f(x) = x^2 - 4x + 3$ , for what value(s) of $x$ does $f(x) = 0$ ?

💡 Show Solution

Step 1: Set the function equal to zero: $x^2 - 4x + 3 = 0$

Step 2: Factor: $(x - 1)(x - 3) = 0$

Step 3: Apply zero product property: $x - 1 = 0 \implies x = 1$ $x - 3 = 0 \implies x = 3$

Check: $f(1) = 1 - 4 + 3 = 0$ ✓ and $f(3) = 9 - 12 + 3 = 0$ ✓

Answer: $x = 1$ and $x = 3$

SAT Context: These are the $x$ -intercepts (zeros/roots) of the function's graph.

10Problem 10medium

❓ Question:

If $f(x) = x^2 - 4x + 3$ , for what value(s) of $x$ does $f(x) = 0$ ?

💡 Show Solution

Step 1: Set the function equal to zero: $x^2 - 4x + 3 = 0$

Step 2: Factor: $(x - 1)(x - 3) = 0$

Step 3: Apply zero product property: $x - 1 = 0 \implies x = 1$ $x - 3 = 0 \implies x = 3$

Check: $f(1) = 1 - 4 + 3 = 0$ ✓ and $f(3) = 9 - 12 + 3 = 0$ ✓

Answer: $x = 1$ and $x = 3$

SAT Context: These are the $x$ -intercepts (zeros/roots) of the function's graph.

11Problem 11medium

❓ Question:

The function $g$ is defined by $g(x) = 2x + 1$ . If $g(a) = 13$ , what is the value of $g(2a)$ ?

💡 Show Solution

Step 1: Find $a$ using $g(a) = 13$ : $2a + 1 = 13$ $2a = 12$ $a = 6$

Step 2: Find $g(2a) = g(12)$ : $g(12) = 2(12) + 1 = 25$

Answer: $g(2a) = 25$

Alternative approach: Notice $g(2a) = 2(2a) + 1 = 4a + 1$ . Since $2a = 12$ , we get $g(2a) = 2(12) + 1 = 25$ .

12Problem 12medium

❓ Question:

The function $g$ is defined by $g(x) = 2x + 1$ . If $g(a) = 13$ , what is the value of $g(2a)$ ?

💡 Show Solution

Step 1: Find $a$ using $g(a) = 13$ : $2a + 1 = 13$ $2a = 12$ $a = 6$

Step 2: Find $g(2a) = g(12)$ : $g(12) = 2(12) + 1 = 25$

Answer: $g(2a) = 25$

Alternative approach: Notice $g(2a) = 2(2a) + 1 = 4a + 1$ . Since $2a = 12$ , we get $g(2a) = 2(12) + 1 = 25$ .

13Problem 13hard

❓ Question:

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

💡 Show Solution

Step 1: Evaluate the inner function first — find $g(2)$ : $g(2) = (2)^2 - 1 = 4 - 1 = 3$

Step 2: Now evaluate $f(g(2)) = f(3)$ : $f(3) = 2(3) + 3 = 9$

Answer: $f(g(2)) = 9$

Key concept: Composition of functions — work from the inside out. First evaluate $g(2)$ , then plug that result into $f$ .

14Problem 14hard

❓ Question:

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

💡 Show Solution

Step 1: Evaluate the inner function first — find $g(2)$ : $g(2) = (2)^2 - 1 = 4 - 1 = 3$

Step 2: Now evaluate $f(g(2)) = f(3)$ : $f(3) = 2(3) + 3 = 9$

Answer: $f(g(2)) = 9$

Key concept: Composition of functions — work from the inside out. First evaluate $g(2)$ , then plug that result into $f$ .

15Problem 15expert

❓ Question:

The graph of $y = f(x)$ passes through the point $(3, 7)$ . If $g(x) = f(2x - 1) + 3$ , what point must lie on the graph of $y = g(x)$ ?

💡 Show Solution

Step 1: We know $f(3) = 7$ (since the graph passes through $(3, 7)$ ).

Step 2: We need to find a value of $x$ where $g(x)$ can be evaluated. For $g(x) = f(2x - 1) + 3$ , we need $2x - 1 = 3$ (so that $f$ is evaluated at 3, which we know).

$2x - 1 = 3$ $2x = 4$ $x = 2$

Step 3: Calculate $g(2)$ : $g(2) = f(2(2) - 1) + 3 = f(3) + 3 = 7 + 3 = 10$

Answer: The point $(2, 10)$ lies on the graph of $y = g(x)$ .

SAT Tip: For transformation questions, figure out what input to $g$ produces a known input to $f$ .

16Problem 16expert

❓ Question:

The graph of $y = f(x)$ passes through the point $(3, 7)$ . If $g(x) = f(2x - 1) + 3$ , what point must lie on the graph of $y = g(x)$ ?

💡 Show Solution

Step 1: We know $f(3) = 7$ (since the graph passes through $(3, 7)$ ).

Step 2: We need to find a value of $x$ where $g(x)$ can be evaluated. For $g(x) = f(2x - 1) + 3$ , we need $2x - 1 = 3$ (so that $f$ is evaluated at 3, which we know).

$2x - 1 = 3$ $2x = 4$ $x = 2$

Step 3: Calculate $g(2)$ : $g(2) = f(2(2) - 1) + 3 = f(3) + 3 = 7 + 3 = 10$

Answer: The point $(2, 10)$ lies on the graph of $y = g(x)$ .

SAT Tip: For transformation questions, figure out what input to $g$ produces a known input to $f$ .

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❓ Frequently Asked Questions

What is Functions?▾

Master function notation, evaluation, and transformations for SAT

How can I study Functions effectively?▾

Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 16 problems provided, checking solutions as you go. Regular review and active practice are key to retention.

Is this Functions study guide free?▾

Yes — all study notes, flashcards, and practice problems for Functions on Study Mondo are free to access. No account is needed.

What course covers Functions?▾

Functions is part of the SAT Prep course on Study Mondo, specifically in the Algebra section. You can explore the full course for more related topics and practice resources.

Are there practice problems for Functions?▾

Yes, this page includes 16 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.

Functions

Try the Interactive Version!

Functions (SAT)

Function Notation

Types of SAT Function Questions

1. Evaluation

2. Composition

3. Finding Input

Domain and Range

SAT Function Tricks

📚 Practice Problems

1Problem 1easy

❓ Question:

2Problem 2easy

❓ Question:

3Problem 3medium

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4Problem 4medium

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5Problem 5hard

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6Problem 6hard

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7Problem 7easy

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8Problem 8easy

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9Problem 9medium

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10Problem 10medium

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11Problem 11medium

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12Problem 12medium

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13Problem 13hard

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14Problem 14hard

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15Problem 15expert

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16Problem 16expert

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Practice Lab

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❓ Frequently Asked Questions