ACT Functions Practice Questions with Answers

Mastering functions is a critical step for any student aiming for a high score on the math section of the ACT. Functions represent the relationship between an input and an output, and they appear in various forms, including linear equations, parabolas, and complex compositions. To excel, you must understand how to evaluate functions, interpret graphs, and manipulate nested expressions. This guide provides a comprehensive overview and plenty of ACT Functions Practice Questions with Answers to help you build confidence.

Concept Explanation

An ACT function is a mathematical rule that assigns exactly one output value, typically denoted as $f(x)$, to each specific input value $x$. Think of a function as a machine: you drop a number into the input slot, the machine performs a specific operation, and it spits out a resulting number. On the ACT, you will encounter several types of function problems, including evaluating functions at specific points, finding the domain (all possible $x$-values) and range (all possible $y$-values), and working with composite functions like $f(g(x))$.

When you see the notation $f(x) = 2x + 3$, it means that for any value you choose for $x$, you must multiply it by 2 and then add 3. For more advanced preparation, you can explore our ACT Prep hub to see how functions integrate with other algebraic concepts. Additionally, understanding how functions behave is foundational for higher-level math found in standard exams like the Khan Academy Algebra sequences. It is also helpful to use tools like an AI Question Generator to create custom drills for specific function types you find challenging.

Key Function Types on the ACT

• Linear Functions: Functions in the form $f(x) = mx + b$, producing a straight line.
• Quadratic Functions: Functions in the form $f(x) = ax^2 + bx + c$, producing a parabola.
• Composite Functions: Where one function is nested inside another, such as $h(x) = f(g(x))$.
• Piecewise Functions: Functions defined by different rules for different intervals of $x$.

Solved Examples

Reviewing these step-by-step solutions will help you understand the logic required for the ACT math section.

Example 1: Evaluating a Function
If $f(x) = 3x^2 - 5x + 2$, what is the value of $f(-2)$?

1. Substitute $-2$ for every instance of $x$ in the equation: $f(-2) = 3(-2)^2 - 5(-2) + 2$.
2. Calculate the squared term: $(-2)^2 = 4$, so the expression becomes $3(4) - 5(-2) + 2$.
3. Perform the multiplications: $12 + 10 + 2$.
4. Add the results: $12 + 10 + 2 = 24$. The answer is 24.

Example 2: Composite Functions
Given $f(x) = 2x + 1$ and $g(x) = x^2$, find $f(g(3))$.

1. Start with the inner function: Find $g(3)$. Since $g(x) = x^2$, $g(3) = 3^2 = 9$.
2. Take the result (9) and plug it into the outer function $f(x)$.
3. Calculate $f(9)$: $f(9) = 2(9) + 1 = 18 + 1 = 19$. The answer is 19.

Example 3: Inverse Function Concepts
If $f(x) = \frac{x-5}{2}$, for what value of $x$ does $f(x) = 10$?

1. Set the entire function equal to 10: $\frac{x-5}{2} = 10$.
2. Multiply both sides by 2 to clear the fraction: $x - 5 = 20$.
3. Add 5 to both sides: $x = 25$. The answer is 25.

Practice Questions

Test your skills with these practice problems. They range from basic evaluation to complex multi-step reasoning.

1. If $f(x) = 4x - 7$, what is $f(5)$?

2. Given $g(x) = \sqrt{x + 9}$, find the value of $g(16)$.

3. Let $h(x) = 2x^2 - 3$. What is the value of $h(h(1))$?

4. If $f(x) = 3x + 2$ and $g(x) = x - 5$, find the expression for $f(g(x))$.

5. The function $p(t) = 100(2)^t$ models the growth of bacteria over $t$ hours. How many bacteria are present after 3 hours?

6. For the function $k(x) = \frac{12}{x-4}$, which value of $x$ is not in the domain?

7. If $f(x) = x^2 + kx + 10$ and $f(2) = 20$, what is the value of $k$?

8. Let $f(x) = |x - 10|$. What is $f(4)$?

9. A function is defined as $f(x) = 5x - 2$. If $f(a) = 13$, what is the value of $a$?

10. Given $g(x) = x^2 - x$, find $g(x+1)$.

Answers & Explanations

1. Answer: 13. Substitute 5 into the function: $f(5) = 4(5) - 7 = 20 - 7 = 13$.
2. Answer: 5. Substitute 16 into the function: $g(16) = \sqrt{16 + 9} = \sqrt{25} = 5$.
3. Answer: -1. First, find $h(1) = 2(1)^2 - 3 = 2 - 3 = -1$. Now, find $h(-1) = 2(-1)^2 - 3 = 2(1) - 3 = -1$.
4. Answer: $3x - 13$. Substitute $g(x)$ into $f(x)$: $f(g(x)) = 3(x - 5) + 2 = 3x - 15 + 2 = 3x - 13$.
5. Answer: 800. Substitute $t = 3$: $p(3) = 100(2)^3 = 100(8) = 800$.
6. Answer: 4. The domain of a rational function excludes values that make the denominator zero. Setting $x - 4 = 0$ gives $x = 4$.
7. Answer: 3. Plug in $x = 2$ and set the result to 20: $2^2 + k(2) + 10 = 20$. This simplifies to $4 + 2k + 10 = 20$, then $14 + 2k = 20$, so $2k = 6$, and $k = 3$.
8. Answer: 6. Substitute 4 into the absolute value function: $f(4) = |4 - 10| = |-6| = 6$.
9. Answer: 3. Set the function equal to 13: $5a - 2 = 13$. Add 2 to both sides to get $5a = 15$, then divide by 5 to get $a = 3$.
10. Answer: $x^2 + x$. Substitute $(x+1)$ for $x$: $(x+1)^2 - (x+1) = (x^2 + 2x + 1) - x - 1 = x^2 + x$.

1. If \( f(x) = 2x^3 \), what is \( f(3) \)?

• A18
• B54
• C27
• D12

Frequently Asked Questions

What is the difference between domain and range?

The domain refers to all possible input values ($x$-values) for which the function is defined, while the range refers to the set of all possible output values ($y$-values) the function can produce. On the ACT, domain restrictions often occur with square roots (cannot be negative) and denominators (cannot be zero).

How do I solve composite functions like $f(g(x))$?

To solve a composite function, you work from the inside out by evaluating the inner function first and then using that result as the input for the outer function. If the problem uses variables, substitute the entire expression of the inner function into every $x$ of the outer function.

Can a function have more than one output for a single input?

No, by definition, a function must assign exactly one output to each input. If a relationship assigns multiple outputs to one input, such as a vertical line or a circle, it fails the "Vertical Line Test" and is not considered a function.

What are piecewise functions on the ACT?

Piecewise functions are functions that use different formulas for different parts of their domain. When evaluating these, you must first look at the given $x$-value to determine which specific rule or "piece" of the function applies to that value.

How do I find the inverse of a function?

To find the inverse of a function, replace $f(x)$ with $y$, swap the $x$ and $y$ variables, and then solve the new equation for $y$. The resulting expression is the inverse function, often denoted as $f^{-1}(x)$.

For students preparing for other rigorous exams, such as pharmacy boards, you might find our NAPLEX Pharmacokinetics Calculation practice or Elimination Rate questions useful for seeing how functions apply to real-world science. If you need more general math help, visit WolframAlpha for computational verification.