SAT II Math II : Range and Domain

Study concepts, example questions & explanations for SAT II Math II

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Example Questions

Example Question #1 : Functions And Graphs

Define \(\displaystyle f(x)= \frac{x- 4}{x-7}\).

Give the range of \(\displaystyle f\).

Possible Answers:

\(\displaystyle (-\infty, 7) \cup (7, \infty)\)

\(\displaystyle (-\infty, 4) \cup (4, \infty)\)

\(\displaystyle (-\infty, 4) \cup (7, \infty)\)

\(\displaystyle (-\infty, 4) \cup (4, 7) \cup (7, \infty)\)

The correct range is not among the other responses.

Correct answer:

The correct range is not among the other responses.

Explanation:

The function can be rewritten as follows:

\(\displaystyle f(x)= \frac{x- 4}{x-7}\)

\(\displaystyle f(x)= \frac{x- 7 + 3}{x-7}\)

\(\displaystyle f(x)= \frac{x- 7 }{x-7}+ \frac{ 3}{x-7}\)

\(\displaystyle f(x)=1+ \frac{ 3}{x-7}\)

The expression \(\displaystyle \frac{ 3}{x-7}\) can assume any value except for 0, so the expression \(\displaystyle f(x)=1+ \frac{ 3}{x-7}\) can assume any value except for 1. The range is therefore the set of all real numbers except for 1, or

\(\displaystyle (-\infty, 1) \cup (1, \infty)\).

This choice is not among the responses.

Example Question #2 : Functions And Graphs

Define \(\displaystyle f(x)= \frac{x- 4}{x-7}\).

Give the domain of \(\displaystyle f\).

Possible Answers:

\(\displaystyle (-\infty, 4) \cup (4, 7) \cup (7, \infty)\)

\(\displaystyle (-\infty, 4) \cup (4, \infty)\)

\(\displaystyle (-\infty, 4) \cup (7, \infty)\)

\(\displaystyle (4,7)\)

\(\displaystyle (-\infty, 7) \cup (7, \infty)\)

Correct answer:

\(\displaystyle (-\infty, 7) \cup (7, \infty)\)

Explanation:

In a rational function, the domain excludes exactly the value(s) of the variable which make the denominator equal to 0. Set the denominator to find these values:

\(\displaystyle x- 7 =0\)

\(\displaystyle x = 7\)

The domain is the set of all real numbers except 7 - that is, \(\displaystyle (-\infty, 7) \cup (7, \infty)\).

Example Question #1 : Range And Domain

Define \(\displaystyle f(x) = \sqrt[3]{100 - x^{2}}\)

Give the domain of \(\displaystyle f\)

Possible Answers:

\(\displaystyle \left [-10, \infty)\)

\(\displaystyle (-\infty,10]\)

\(\displaystyle (-\infty, \infty)\)

\(\displaystyle \left [0, 10 \right ]\)

\(\displaystyle \left [-10, 10 \right ]\)

Correct answer:

\(\displaystyle (-\infty, \infty)\)

Explanation:

Every real number has one real cube root, so there are no restrictions on the radicand of a cube root expression. The domain is the set of all real numbers.

Example Question #1 : Properties Of Functions And Graphs

Define \(\displaystyle f(x) = 2- 4\cos x\)

Give the range of \(\displaystyle f\).

Possible Answers:

\(\displaystyle \left ( -\infty, \infty\right )\)

\(\displaystyle [2,6]\)

\(\displaystyle [-6, -2]\)

\(\displaystyle [-2,6]\)

\(\displaystyle [-6, 2]\)

Correct answer:

\(\displaystyle [-2,6]\)

Explanation:

\(\displaystyle -1 \leq \cos x \leq 1\) for any real value of \(\displaystyle x\).

Therefore,

\(\displaystyle -4(-1) \cdot \geq -4 \cdot \cos x \geq -4 \cdot 1\)

\(\displaystyle 4 \geq -4 \cos x \geq -4\)

\(\displaystyle 4 + 2 \geq -4 \cos x + 2 \geq -4 + 2\)

\(\displaystyle 6 \geq 2-4 \cos x \geq -2\)

\(\displaystyle -2 \leq f(x) \leq 6\)

The range is \(\displaystyle [-2,6]\).

Example Question #1 : Range And Domain

Define \(\displaystyle f(x) = 2- \cos 4x\)

Give the range of \(\displaystyle f\).

Possible Answers:

\(\displaystyle \left [ 1,3\right ]\)

\(\displaystyle \left [ 1,2\right ]\)

\(\displaystyle [-2,6]\)

\(\displaystyle \left ( -\infty, \infty\right )\)

\(\displaystyle [-2,2]\)

Correct answer:

\(\displaystyle \left [ 1,3\right ]\)

Explanation:

\(\displaystyle -1 \leq \cos \theta \leq 1\) for any real value of \(\displaystyle \theta\), so

\(\displaystyle -1 \leq \cos 4x \leq 1\)

\(\displaystyle -1 \cdot \left (-1 \right ) \geq -1 \cdot \cos 4x \geq -1 \cdot 1\)

\(\displaystyle 1 \geq - \cos 4x \geq -1\)

\(\displaystyle 1+ 2 \geq - \cos 4x + 2 \geq -1 + 2\)

\(\displaystyle 3 \geq 2 - \cos 4x \geq 1\)

\(\displaystyle 1 \leq f(x) \leq 3\),

making the range \(\displaystyle \left [ 1,3\right ]\).

Example Question #3 : Properties Of Functions And Graphs

Define \(\displaystyle f (x) = \sqrt{16-x^{2}}\).

Give the range of \(\displaystyle f\)

Possible Answers:

\(\displaystyle (-\infty, 4]\)

\(\displaystyle [0, 16]\)

\(\displaystyle [-16,16]\)

\(\displaystyle [0, 4]\)

\(\displaystyle [-4,4]\)

Correct answer:

\(\displaystyle [0, 4]\)

Explanation:

The radicand within a square root symbol must be nonnegative, so

\(\displaystyle 16 - x^{2} \geq 0\)

\(\displaystyle 16 \geq x^{2}\)

\(\displaystyle x^{2} \leq 16\)

This happens if and only if \(\displaystyle -4 \leq x \leq 4\), so the domain of \(\displaystyle f\) is \(\displaystyle [-4,4]\).

 

\(\displaystyle f (x) = \sqrt{16-x^{2}}\) assumes its greatest value when \(\displaystyle 16-x^{2}\), which is the point on \(\displaystyle [-4,4]\) where \(\displaystyle x^{2}\) is least - this is at \(\displaystyle x = 0\).

\(\displaystyle f (0) = \sqrt{16-0^{2}} = \sqrt{16} = 4\)

 

 

Similarly, \(\displaystyle f (x) = \sqrt{16-x^{2}}\) assumes its least value when \(\displaystyle 16-x^{2}\), which is the point on \(\displaystyle [-4,4]\) where \(\displaystyle x^{2}\) is greatest - this is at \(\displaystyle x = \pm 4\).

\(\displaystyle f (4) = \sqrt{16-4^{2}} = \sqrt{0} = 0\)

\(\displaystyle f (-4) = \sqrt{16- \left (-4 \right )^{2}} = \sqrt{0} = 0\)

 

Therefore, the range of \(\displaystyle f\) is \(\displaystyle [0, 4]\).

Example Question #1 : Functions And Graphs

Define \(\displaystyle f(x) = \frac{5}{\sin 4x}\)

Give the range of \(\displaystyle f\).

Possible Answers:

\(\displaystyle ( -5 , 5 )\)

\(\displaystyle \left ( -\frac{5}{4}, \frac{5}{4} \right )\)

\(\displaystyle (- \infty, -5] \cup [5, \infty)\)

\(\displaystyle (- \infty, 0) \cup (0, \infty)\)

\(\displaystyle \left (- \infty, -\frac{5}{4} \right )\cup\left ( \frac{5}{4}, \infty \right )\)

Correct answer:

\(\displaystyle (- \infty, -5] \cup [5, \infty)\)

Explanation:

\(\displaystyle f(x) = \frac{5}{\sin 4x}\) can be rewritten as \(\displaystyle f (x) = 5 \csc 4x\).

For all real values of \(\displaystyle \theta\),

\(\displaystyle \csc \theta\leq -1\) or \(\displaystyle \csc \theta\geq 1\).

Therefore,

\(\displaystyle \csc 4x \leq -1\) or \(\displaystyle \csc 4x\geq 1\) and

\(\displaystyle 5 \csc 4x \leq -5\) or \(\displaystyle 5 \csc 4x\geq 5\) .

The range of \(\displaystyle f\) is \(\displaystyle (- \infty, -5] \cup [5, \infty)\).

Example Question #2 : Properties Of Functions And Graphs

What is the domain of the function \(\displaystyle f(x)=x^2+3?\)

Possible Answers:

\(\displaystyle (3,\infty)\)

\(\displaystyle (-\infty ,\infty)\)

\(\displaystyle [-\infty ,\infty]\)

\(\displaystyle [3,\infty)\)

\(\displaystyle (-3,\infty )\)

Correct answer:

\(\displaystyle (-\infty ,\infty)\)

Explanation:

The domain of a function is all the x-values that in that function. The function \(\displaystyle f(x)=x^2+3\) is a upward facing parabola with a vertex as (0,3). The parabola keeps getting wider and is not bounded by any x-values so it will continue forever. Parenthesis are used because infinity is not a definable number and so it can not be included.

Example Question #1 : Functions And Graphs

What is the domain of the function?   \(\displaystyle y=\sqrt{-10x+3}\)

Possible Answers:

\(\displaystyle x\geq \frac{3}{10}\)

\(\displaystyle x\leq- \frac{3}{10}\)

\(\displaystyle \textup{All real numbers.}\)

\(\displaystyle x\leq \frac{3}{10}\)

\(\displaystyle -\frac{3}{10}\leq x\leq \frac{3}{10}\)

Correct answer:

\(\displaystyle x\leq \frac{3}{10}\)

Explanation:

Notice this function resembles the parent function \(\displaystyle y=\sqrt {x}\).  The value of \(\displaystyle x\) must be zero or greater.

Set up an inequality to determine the domain of \(\displaystyle x\).

\(\displaystyle -10x+3\geq 0\)

Subtract three from both sides.

\(\displaystyle -10x+3-3\geq 0-3\)

\(\displaystyle -10x\geq -3\)

Divide by negative ten on both sides.  The sign will switch.

\(\displaystyle \frac{-10x}{-10}\geq \frac{-3}{-10}\)

The domain is:  \(\displaystyle x\leq \frac{3}{10}\)

Example Question #1 : Range And Domain

What is the range of the function \(\displaystyle f(x)=\frac{2}{x}-3\)?

Possible Answers:

All real numbers except \(\displaystyle 0\).

All real numbers except \(\displaystyle -3\).

All real numbers.

All real numbers except \(\displaystyle 2\).

Correct answer:

All real numbers except \(\displaystyle -3\).

Explanation:

Start by considering the term \(\displaystyle \frac{2}{x}\)\(\displaystyle \frac{2}{x}\) will hold for all values of \(\displaystyle x\), except when \(\displaystyle x=0\). Thus, \(\displaystyle \frac{2}{x}-3\) must be defined by all values except \(\displaystyle -3\) since the equation is just shifted down by \(\displaystyle 3\).

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