SAT Functions
To solve problems about functions on the SAT, you need to know:
Contents
Examples
\[\begin{array}{|c|c|} \hline x & f(x)\\ \hline 3 & -2\\ \hline 4 & -3\\ \hline 5 & -5\\ \hline 6 &\ \ \ 4\\ \hline 7 &\ \ \ 6\\ \hline 8 & -8\\ \hline \end{array}\]
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) \(\ \ 8\)
Correct Answer: E
Solution:
\(g(3)=2h(3)=2f(3+3)=2(f(6))=2\cdot 4 = 8\)
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) \(\ \ 9\)
Correct Answer: D
Solution:
As shown below, \(h(6)=9.\) Therefore, \(a=9\) and \(h(a-6)=h(9-6)=h(3)=4.\)
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) \(\ \ 0 < x < 5\)
Review
If you thought these examples difficult and you need to review the material, these links will help:
SAT Tips for Functions
- 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