GMAT Math : Graphing

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Example Questions

Example Question #1 : Graphing A Function

What is the domain of $y=-2\sqrt{x}$ ?

Possible Answers:

x>0

$x\geq 0$

x=0

$x\leq 0$

x<-2

Correct answer:

$x\geq 0$

Explanation:

To find the domain, we need to decide which values can take. The is under a square root sign, so cannot be negative. can, however, be 0, because we can take the square root of zero. Therefore the domain is $x\geq 0$ .

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Example Question #61 : Graphing

What is the domain of the function $y=\sqrt{4-x^{2}}$ ?

Possible Answers:

$\left ( -2,2 \right )$

$x\leq -2$

$\left [ -2,2 \right ]$

x<-2

x>-2

Correct answer:

$\left [ -2,2 \right ]$

Explanation:

To find the domain, we must find the interval on which $\sqrt{4-x^{2}}$ is defined. We know that the expression under the radical must be positive or 0, so $\sqrt{4-x^{2}}$ is defined when $x^{2}\leq 4$ . This occurs when $x \geq -2$ and $x \leq 2$ . In interval notation, the domain is $\left [ -2,2 \right ]$ .

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Example Question #1 : Graphing A Function

Define the functions and as follows:

$\small \small f (x) = 2x + \sqrt{x - 1}$

$\small g (x) = 6x - \sqrt{x-1}$

What is the domain of the function (f+g)(x) ?

Possible Answers:

$\small \small [1, \infty )$

$\small (-\infty ,\infty )$

$\small (-\infty , 1) \cup (1, \infty )$

$\small \small (-\infty , -1] \cup [1, \infty )$

$\small (1, \infty )$

Correct answer:

$\small \small [1, \infty )$

Explanation:

The domain of (f+g) is the intersection of the domains of and . and are each restricted to all values of that allow the radicand x-1 to be nonnegative - that is,

$\small \small x - 1\geq 0$ , or

$\small \small \small x \geq 1$

Since the domains of and are the same, the domain of (f+g) is also the same. In interval form the domain of (f+g) is $\small \small [1, \infty )$

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Example Question #4 : How To Graph A Function

Define $g (x) = \frac{1}{\sqrt[3]{x -27}}$

What is the natural domain of ?

Possible Answers:

$(-\infty , - 27)\cup (27, \infty )$

$(-\infty , - 3)\cup (3, \infty )$

$(-\infty , 27)\cup (27, \infty )$

$(3, \infty )$

$(27, \infty )$

Correct answer:

$(-\infty , 27)\cup (27, \infty )$

Explanation:

The radical in and of itself does not restrict the domain, since every real number has a real cube root. However, since the expression $\sqrt[3]{x -27}$ is in a denominator, it cannot be equal to zero, so the domain excludes the value(s) for which

$\sqrt[3]{x -27} = 0$

$\left (\sqrt[3]{x -27} \right )^{3}= 0^{3}$

x -27 = 0

x = 27

27 is the only number excluded from the domain.

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Example Question #5 : Graphing A Function

Define $g(x) = \frac{x}{x^{2}+3x-4 }$

What is the natural domain of ?

Possible Answers:

$( - \infty ,-1 ) \cup (-1,4) \cup (4, \infty )$

$( - \infty ,-1 ) \cup (-1,0) \cup (0,4) \cup (4, \infty )$

$( - \infty ,-4 ) \cup (-4,0) \cup (0,1) \cup (1, \infty )$

$( - \infty ,-4 ) \cup (-4,1) \cup (1, \infty )$

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

Correct answer:

$( - \infty ,-4 ) \cup (-4,1) \cup (1, \infty )$

Explanation:

Since the expression $x^{2}+3x-4$ is in a denominator, it cannot be equal to zero, so the domain excludes the value(s) for which $x^{2}+3x-4 =0$ . We solve for by factoring the polynomial, which we can do as follows:

$x^{2}+3x-4 =0$

(x+?)(x+?)=0

Replacing the question marks with integers whose product is and whose sum is 3:

(x+4)(x-1)=0

$x + 4 = 0 \Rightarrow x = -4$

$x -1 = 0 \Rightarrow x = 1$

Therefore, the domain excludes these two values of .

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Example Question #2 : How To Graph A Function

Define $f (x) = \frac{1}{x ^{2}- 25 }$ .

What is the natural domain of ?

Possible Answers:

$(-\infty , -5 ) \cup (5, \infty)$

$(-\infty , -5 ) \cup \left ( -5,5\right ) \cup (5, \infty)$

$(-\infty , -5 ) \cup \left ( -5, \infty)$

(-5 , 5)

$(-\infty , 5 ) \cup \left ( 5, \infty)$

Correct answer:

$(-\infty , -5 ) \cup \left ( -5,5\right ) \cup (5, \infty)$

Explanation:

The only restriction on the domain of is that the denominator cannot be 0. We set the denominator to 0 and solve for to find the excluded values:

$x ^{2}-25 =0$

$x ^{2}= 25$

$x = -5 \textrm{ or }x = 5$

The domain is the set of all real numbers except those two - that is,

$(-\infty , -5 ) \cup \left ( -5,5\right ) \cup (5, \infty)$ .

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Example Question #1 : Coordinate Geometry

What is the -intercept of the graph of $y = \log _{2}(x+ 4)$ ?

Possible Answers:

The graph has no -intercept.

(-3,0)

(1,0)

(2,0)

(-4,0)

Correct answer:

(-3,0)

Explanation:

Set y = 0 and solve:

$y = \log _{2}(x+ 4)$

$\log _{2}(x+ 4) = 0$

$2 ^ {\log _{2}(x+ 4) }= 2 ^ { 0}$

x + 4 = 1

x + 4 - 4 = 1 - 4

x = -3

The -intercept is (-3,0) .

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Example Question #1 : Coordinate Geometry

What is the -intercept of the graph of $y = \log _{3}(x+ 9)$ ?

Possible Answers:

(0,2)

The graph has no -intercept.

(0,3)

(0,-6)

(0,-9)

Correct answer:

(0,2)

Explanation:

Set x= 0 and evaluate :

$y = \log _{3}(x+ 9)$

$y = \log _{3}(0+ 9) = \log _{3}9$

Since $3 ^{2} =9$ ,

$\log _{3}9 =2$ , and the -intercept is (0,2) .

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Example Question #1 : Coordinate Geometry

What is the vertical asymptote of the graph of $y = \log_{5} (x-7)$ ?

Possible Answers:

The graph has no vertical asymptote.

x = 12

x = 6

x = 7

x = 8

Correct answer:

x = 7

Explanation:

The graph of a logarithmic function has a vertical asymptote which can be found by finding the value at which the power is equal to 0:

$x-7 = 0 \Rightarrow x = 7$

If $x \leq 7$ , then $\log_{5} (x-7)$ is an undefined expression, so the vertical asymptote is x = 7 .

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Example Question #1 : Graphing A Logarithm

Define a function as follows:

$g(x) =2 \log_{4}(x-4)+ 3$

Give the -intercept of the graph of .

Possible Answers:

The graph of has no -intercept.

$\left ( 3\frac{7}{8}, 0 \right )$

$\left ( 4\frac{1}{8}, 0 \right )$

(12,0)

(-4, 0)

Correct answer:

$\left ( 4\frac{1}{8}, 0 \right )$

Explanation:

Set g(x) =0 and evaluate to find the -coordinate of the -intercept.

$g(x) =2 \log_{4}(x-4)+ 3 = 0$

$2 \log_{4}(a-4) = -3$

$\log_{4}(a-4) = -\frac{3}{2}$

This can be rewritten in exponential form:

$a - 4 = 4 ^{-\frac{3}{2}}= \frac{1}{4 ^{ \frac{3}{2}}} = \frac{1}{(\sqrt{4 })^{ 3}} = \frac{1}{2^{ 3}} = \frac{1}{8}$

$a = 4\frac{1}{8}$

The -intercept of the graph of is $\left ( 4\frac{1}{8}, 0 \right )$ .

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GMAT Math : Graphing

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Graphing A Function

Example Question #61 : Graphing

Example Question #1 : Graphing A Function

Example Question #4 : How To Graph A Function

Example Question #5 : Graphing A Function

Example Question #2 : How To Graph A Function

Example Question #1 : Coordinate Geometry

Example Question #1 : Coordinate Geometry

Example Question #1 : Coordinate Geometry

Example Question #1 : Graphing A Logarithm

All GMAT Math Resources

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