SAT Functions

To solve problems about functions on the SAT, you need to know:

• function terminology
• how to evaluate functions
• function arithmetic
• function composition
• how to interpret graphs of functions

$\begin{array}{|c|c|} \hline x & f(x)\\ \hline 3 & -2\\ \hline 4 & -3\\ \hline 5 & -5\\ \hline 6 &\ \ \ 4\\ \hline 7 &\ \ \ 6\\ \hline 8 & -8\\ \hline \end{array}$

The table above shows some values of the function $f(x).$ If $h(x)=f(x+3)$ and $g(x)=2h(x)$, what is $g(3)$?

(A) $\ \ -4$
(B) $\ \ -3$
(C) $\ \ -2$
(D) $\ \ 4$
(E) $\ \ 8$

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Correct Answer: E

Solution:

$g(3)=2h(3)=2f(3+3)=2(f(6))=2\cdot 4 = 8$

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Incorrect Choices:

(A)
Tip: Read the entire question carefully.
You will get this wrong answer if you do this: $g(3)=2h(3)=2f(3).$ We are told that $h(x)=f(x+3),$ not that, $h(x)=f(x).$

(B)
Tip: Just because a number appears in the question doesn’t mean it is the answer.

(C)
Tip: Read the entire question carefully.
If you solve for $f(3)$, you will get this wrong answer.

(D)
Tip: Read the entire question carefully.
If you solve for $h(3)$ instead of $g(3),$ you will get this wrong answer.

The graph of $h(x)$ is shown above. If $h(6)=a$, which of the following is $h(a-6)$?

(A) $\ \ 0$
(B) $\ \ 1$
(C) $\ \ 3$
(D) $\ \ 4$
(E) $\ \ 9$

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Correct Answer: D

Solution:

As shown below, $h(6)=9.$ Therefore, $a=9$ and $h(a-6)=h(9-6)=h(3)=4.$

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Incorrect Choices:

(A)
Tip: The line $y = b$ is a horizontal line that crosses the $y$-axis at $(0,b).$
You will get this wrong answer if you find the slope of the horizontal line between $x=2$ and $x=4,$ instead of evaluating $h(x)$ at $3.$

(B)
If you solve for $h(6-6)=h(0)$, you will get this wrong answer. $h(6)=a$ does not mean that $a=6.$ It means that the value of $h(x)$ when $x=6$ is $a$.

(C)
Tip: Read the entire question carefully.
If you find that $h(6)=9$ and you then solve for $h(9)$, instead of $h(9-6),$ you will get this wrong answer.

(E)
Tip: Read the entire question carefully.
If you solve for $h(6)=a$, you will get this wrong answer. We're looking for $h(a-6).$

If you thought these examples difficult and you need to review the material, these links will help:

• Function Terminology
• Evaluating Functions
• Function Arithmetic
• Function Composition
• Interpreting Graphs of Functions
• SAT Linear Functions
• SAT Quadratic Functions
• SAT Absolute Value

• The line $y = b$ is a horizontal line that crosses the $y$-axis at $(0,b).$
• The line $x = a$ is a vertical line that crosses the x-axis at $(a,0).$
• For $f(x)=\sqrt{x}, \quad \text{Domain:}\ x\geq 0; \quad \text{Range:}\ f(x) \geq 0.$
• The domain of $f(g(x))$ is the set of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.
• $\sqrt{x^{2}} = \begin{cases} -x &\mbox{if } x < 0 \\ x & \mbox{if } x \geq 0. \\ \end{cases}$
• $|x| = \begin{cases} -x &\mbox{if } x < 0 \\ x & \mbox{if } x \geq 0. \\ \end{cases}$
• Follow order of operations.
• SAT General Tips