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 \(h(a-6).\)

The function \(f\) is shown above. What are all the values of \(x\) for which \(f\) is negative?

(A) \(\ \ -\infty < x< -3\ \text{and}\ 0 < x < 5\)
(B) \(\ \ -\infty < x < -2\)
(C) \(\ \ -3 < x <0\ \text{and}\ 5 < x < \infty\)
(D) \(\ \ -3 < x < 0\)
(E) \(\ \ 0 < x < 5\)

A B C D E

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