Medium SAT Functions Practice Questions

Medium SAT Functions Practice Questions

Mastering function notation, transformations, and algebraic manipulation is essential for scoring high on the math section of the SAT. This guide provides comprehensive Medium SAT Functions Practice Questions to help you bridge the gap between basic concepts and advanced problem-solving.

Concept Explanation

SAT functions are mathematical relationships where each input value $x$ (the domain) corresponds to exactly one output value $f(x)$ (the range). At the medium difficulty level, the SAT moves beyond simple substitution and requires students to understand composite functions, graph transformations, and interpreting functions within real-world contexts. For example, if you are given $f(x) = 2x + 3$, finding $f(g(x))$ involves replacing every $x$ in the first function with the entire expression of the second. Understanding how to manipulate these expressions is a core skill often found in Medium SAT Math Practice Questions. Furthermore, you must be comfortable with the coordinate plane, where $y = f(x)$, and recognize that the roots or zeros of a function are the $x$-intercepts where $f(x) = 0$. For more foundational review, you might also look at Medium SAT Algebra Practice Questions.

Solved Examples

Below are fully worked examples demonstrating the logic required for medium-level function problems.

1. Example 1: Function Substitution
   If $f(x) = x^2 - 5x + 4$, what is the value of $f(x-3)$?
   1. Substitute $(x-3)$ for every $x$ in the original function: $f(x-3) = (x-3)^2 - 5(x-3) + 4$.
   2. Expand the squared term: $(x-3)(x-3) = x^2 - 6x + 9$.
   3. Distribute the -5: $-5x + 15$.
   4. Combine all terms: $x^2 - 6x + 9 - 5x + 15 + 4$.
   5. Simplify: $x^2 - 11x + 28$.
2. Example 2: Composite Functions
   Given $g(x) = 3x - 2$ and $h(x) = \frac{x}{2} + 4$, find the value of $g(h(6))$.
   1. Start with the inner function: Calculate $h(6)$.
   2. $h(6) = \frac{6}{2} + 4 = 3 + 4 = 7$.
   3. Now use the result as the input for $g(x)$: Find $g(7)$.
   4. $g(7) = 3(7) - 2 = 21 - 2 = 19$.
3. Example 3: Interpreting Linear Functions
   The function $C(p) = 15p + 200$ represents the total cost in dollars to produce $p$ phones. What does the value 15 represent in this context?
   1. Identify the form of the function: This is a linear function $y = mx + b$.
   2. Recall that $m$ (the coefficient of the variable) represents the rate of change.
   3. In this context, for every additional phone produced ($p$), the cost increases by $15.
   4. Therefore, 15 is the cost per phone.

Practice Questions

Test your skills with these Medium SAT Functions Practice Questions. Ensure you show your work for each step.

1. If $f(x) = \frac{2x+6}{x-1}$, what is the value of $f(5)$?

2. The graph of the function $g$ is a parabola in the $xy$-plane. If the vertex of the parabola is $(3, -4)$, which of the following could be the equation for $g(x)$?
A) $g(x) = (x+3)^2 - 4$
B) $g(x) = (x-3)^2 - 4$
C) $g(x) = (x-3)^2 + 4$
D) $g(x) = (x+3)^2 + 4$

3. Let $f(x) = 4x^2 - kx + 2$. If $f(2) = 10$, what is the value of $k$?

4. If $h(x) = 2^x + 5$, what is the value of $h(3) - h(1)$?

5. A function $q(x)$ is defined such that $q(x) = f(x) + g(x)$. If $f(x) = 3x^2 - 4$ and $g(x) = 2x + 1$, what is $q(-2)$?

6. The function $f$ is defined by $f(x) = ax^2 + 7$. For the constant $a$, $f(3) = 34$. What is the value of $f(-3)$?

7. Which of the following functions has a graph that does not intersect the $x$-axis?
A) $f(x) = x^2 - 4$
B) $f(x) = x^2 + 4$
C) $f(x) = (x-4)^2$
D) $f(x) = -x^2 + 4$

8. If $g(x) = 2x - 1$, find the value of $x$ such that $g(x+2) = 11$.

9. The function $w(t) = 100(1.05)^t$ models the population of a certain species over $t$ years. By what percentage does the population increase each year?

10. For the function $p(x) = |x - 5| + 2$, what is the minimum value of the function?

Answers & Explanations

1. Answer: 4
   Substitute 5 into the function: $f(5) = \frac{2(5)+6}{5-1} = \frac{10+6}{4} = \frac{16}{4} = 4$.
2. Answer: B
   The vertex form of a parabola is $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex. For vertex $(3, -4)$, the equation must be $g(x) = a(x-3)^2 - 4$. Choice B matches this form.
3. Answer: 4
   Plug in $x=2$ and set the result to 10: $4(2)^2 - k(2) + 2 = 10$. This simplifies to $16 - 2k + 2 = 10$, then $18 - 2k = 10$. Subtracting 18 gives $-2k = -8$, so $k = 4$.
4. Answer: 6
   First find $h(3) = 2^3 + 5 = 8 + 5 = 13$. Then find $h(1) = 2^1 + 5 = 2 + 5 = 7$. Finally, $13 - 7 = 6$.
5. Answer: 5
   First, find $f(-2) = 3(-2)^2 - 4 = 3(4) - 4 = 8$. Next, find $g(-2) = 2(-2) + 1 = -3$. Then $q(-2) = 8 + (-3) = 5$.
6. Answer: 34
   In the function $f(x) = ax^2 + 7$, the variable $x$ is squared. Since $(3)^2 = 9$ and $(-3)^2 = 9$, the output will be the same for both inputs. Therefore, $f(-3) = f(3) = 34$.
7. Answer: B
   A graph does not intersect the $x$-axis if $f(x) = 0$ has no real solutions. For $x^2 + 4 = 0$, $x^2 = -4$, which has no real square root. Visually, this is a parabola shifted up 4 units.
8. Answer: 4
   Set up the equation: $2(x+2) - 1 = 11$. Distribute: $2x + 4 - 1 = 11$, which simplifies to $2x + 3 = 11$. Subtract 3 to get $2x = 8$, so $x = 4$.
9. Answer: 5%
   In an exponential growth function $y = a(1+r)^t$, $r$ represents the growth rate. Here, $1+r = 1.05$, so $r = 0.05$, which is 5%.
10. Answer: 2
    The absolute value expression $|x - 5|$ is always greater than or equal to 0. Its minimum value is 0 (when $x=5$). Therefore, the minimum value of the entire function is $0 + 2 = 2$.

Frequently Asked Questions

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

$f(x) + k$ represents a vertical shift of the graph up or down by $k$ units. In contrast, $f(x + k)$ represents a horizontal shift left or right by $k$ units.

How do I find the zeros of a function on the SAT?

To find the zeros, set the function $f(x)$ equal to zero and solve for $x$. These values represent the points where the graph crosses the horizontal axis.

What are composite functions?

Composite functions, written as $f(g(x))$, involve using the output of one function as the input for another. You evaluate the inner function first and then apply the outer function to that result.

How can I identify a function from a table?

A table represents a function if every unique input in the $x$-column corresponds to exactly one value in the $y$-column. If one $x$ value has two different $y$ values, it is not a function.

What is the vertex form of a quadratic function?

The vertex form is $f(x) = a(x-h)^2 + k$. This form is highly useful on the SAT because it directly tells you the coordinates of the maximum or minimum point, which is $(h, k)$.

Why are functions important for the SAT?

Functions account for a significant portion of the Heart of Algebra and Passport to Advanced Math sections. Mastery of functions is required to solve modeling problems and interpret complex data sets. For additional practice, explore SAT Algebra Practice Questions with Answers.