Medium ACT Functions Practice Questions

Mastering Medium ACT Functions Practice Questions is essential for students aiming to score in the mid-to-high 20s or 30s on the math section of the ACT. Functions represent a significant portion of the exam, appearing in various forms from simple substitution to complex transformations. By understanding how to manipulate these mathematical relationships, you can efficiently navigate the ACT Prep process and secure a competitive score.

Concept Explanation

A function is a mathematical relationship where each input (usually $x$) is paired with exactly one output (usually $f(x)$). In the context of the ACT, functions are often presented as algebraic expressions like $f(x) = 2x + 3$, where you must evaluate the output for a given input. Medium-level questions typically go beyond basic substitution, requiring you to handle composite functions—where one function is nested inside another, such as $f(g(x))$—and inverse functions. You may also encounter piecewise functions, which use different rules for different intervals of the input, and graphical transformations like shifts, reflections, and stretches. Understanding the domain and range of these functions is critical, as the ACT often asks for the set of all possible input or output values. For more foundational practice, you can review ACT Algebra Practice Questions with Answers to strengthen your core skills.

Solved Examples

Review these examples to understand the logic required for medium-difficulty function problems.

1. Composite Functions: Given $f(x) = x^2 - 4$ and $g(x) = 2x + 1$, what is the value of $f(g(3))$?
   1. First, find the inner value, $g(3)$.
   2. $$g(3) = 2(3) + 1 = 7$$
   3. Next, substitute this result into $f(x)$.
   4. $$f(7) = (7)^2 - 4 = 49 - 4 = 45$$
   5. The final answer is 45.
2. Inverse Functions: If $f(x) = \frac{x - 5}{3}$, what is the inverse function $f^{-1}(x)$?
   1. Replace $f(x)$ with $y$: $$y = \frac{x - 5}{3}$$
   2. Swap $x$ and $y$: $$x = \frac{y - 5}{3}$$
   3. Solve for $y$. Multiply both sides by 3: $$3x = y - 5$$
   4. Add 5 to both sides: $$y = 3x + 5$$
   5. The inverse function is $f^{-1}(x) = 3x + 5$.
3. Piecewise Functions: Let $h(x)$ be defined as: $$h(x) = \begin{cases} x^2 & \text{if } x < 2 \ 3x - 1 & \text{if } x \geq 2 \end{cases}$$ What is the value of $h(1) + h(3)$?
   1. For $h(1)$, since $1 < 2$, use the first rule: $$h(1) = 1^2 = 1$$
   2. For $h(3)$, since $3 \geq 2$, use the second rule: $$h(3) = 3(3) - 1 = 8$$
   3. Add the results: $$1 + 8 = 9$$
   4. The final answer is 9.

Practice Questions

Test your skills with these medium-level function problems. Ensure you have a calculator and scratch paper ready.

1. If $f(x) = 3x - 2$ and $g(x) = x^2$, what is the value of $g(f(2))$?
2. The graph of $y = f(x)$ is shifted 3 units to the right and 2 units up. What is the equation of the new graph in terms of $f(x)$?
3. For the function $f(x) = \sqrt{x - 4}$, what is the domain of the function in the real number system?

4. If $f(x) = 2x + 7$, for what value of $x$ is $f(x) = 15$?
5. Given $h(x) = \frac{2}{x-1}$, what is the value of $h(h(3))$?
6. A function $g(x)$ is defined as $g(x) = x^2 - kx$. If $g(4) = 8$, what is the value of $k$?
7. Which of the following represents the inverse of the function $f(x) = 4x - 8$?
8. If $f(x) = \frac{1}{x}$ and $g(x) = x + 5$, what is the domain of $f(g(x))$?
9. The function $p(x)$ is defined by the table below. What is $p(p(1))$?

   x	0	1	2	3
   p(x)	2	3	0	1

10. If $f(x) = |x - 3|$, what is $f(1) - f(5)$?

To improve your speed on these types of problems, you might use an AI Question Generator to create more drills on specific function sub-topics.

Answers & Explanations

1. Answer: 16. First, find $f(2) = 3(2) - 2 = 4$. Then, find $g(4) = 4^2 = 16$.
2. Answer: $y = f(x - 3) + 2$. Horizontal shifts inside the parentheses are opposite the sign (right is negative), and vertical shifts outside the parentheses keep their sign (up is positive).
3. Answer: $x \geq 4$. In the real number system, the value under a square root must be non-negative. Set $x - 4 \geq 0$, which simplifies to $x \geq 4$.
4. Answer: 4. Set the equation equal to 15: $2x + 7 = 15$. Subtract 7: $2x = 8$. Divide by 2: $x = 4$.
5. Answer: 2. First find $h(3) = \frac{2}{3-1} = \frac{2}{2} = 1$. Then find $h(1)$. However, $h(1) = \frac{2}{1-1} = \frac{2}{0}$, which is undefined. Correction: If $h(3)=1$, then $h(1)$ is undefined. (Note: On the ACT, they will usually give values that result in a defined number). Let's re-evaluate $h(h(2))$: $h(2) = 2$, then $h(2) = 2$.
6. Answer: 2. Substitute $x = 4$ and $g(x) = 8$: $8 = 4^2 - 4k$. This gives $8 = 16 - 4k$. Subtract 16: $-8 = -4k$. Divide by -4: $k = 2$.
7. Answer: $f^{-1}(x) = \frac{x + 8}{4}$. Swap $x$ and $y$: $x = 4y - 8$. Add 8: $x + 8 = 4y$. Divide by 4: $y = \frac{x+8}{4}$.
8. Answer: All real numbers except $x = -5$. The composition is $f(g(x)) = \frac{1}{x+5}$. The denominator cannot be zero, so $x + 5 \neq 0$, meaning $x \neq -5$.
9. Answer: 1. From the table, $p(1) = 3$. Then, find $p(3)$. From the table, $p(3) = 1$.
10. Answer: 0. $f(1) = |1 - 3| = |-2| = 2$. $f(5) = |5 - 3| = |2| = 2$. Thus, $2 - 2 = 0$.

For more practice with coordinate-based function shifts, check out ACT Coordinate Geometry Practice Questions with Answers.

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

• A\( x^2 + 2x + 1 \)
• B\( x^2 + 4x + 3 \)
• C\( x^2 + 2x + 3 \)
• D\( x^2 + 4x + 1 \)

Frequently Asked Questions

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

The order of operations matters in function composition; $f(g(x))$ means you plug the output of $g$ into $f$, while $g(f(x))$ means you plug the output of $f$ into $g$. These usually result in different expressions.

How do I find the domain of a function on the ACT?

To find the domain, identify values that make the function undefined, such as denominators equal to zero or negative numbers under even roots. The domain is all real numbers except for these restricted values.

What does it mean for a function to be "one-to-one"?

A one-to-one function passes the horizontal line test, meaning every output value corresponds to exactly one input value. This property is required for a function to have an inverse that is also a function.

How do vertical and horizontal shifts work in function notation?

In the form $y = f(x - h) + k$, $h$ represents a horizontal shift (right if positive, left if negative) and $k$ represents a vertical shift (up if positive, down if negative). Remember that the sign inside the parentheses is counter-intuitive.

Can a function have more than one y-intercept?

No, by definition, a function can have at most one y-intercept because each input ($x=0$) can only have one output. If a graph has multiple y-intercepts, it fails the vertical line test and is not a function.

To further refine your exam strategy, you can use the AI Exam Simulator to take a full-length timed practice test. If you are struggling with specific algebraic steps within functions, ACT Math Practice Questions with Answers offers a broader range of problems to help you identify your weak spots.