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Which statement describes the domain of the function [tex]f(x) = \frac{3x}{4x^2 - 4}[/tex]?

A. all real numbers
B. all nonzero real numbers
C. all real numbers except [tex]x = \frac{3}{4}[/tex]
D. all real numbers except [tex]x = -1[/tex] and [tex]x = 1[/tex]



Answer :

GinnyAnswer
To determine the domain of the function [tex]\( f(x) = \frac{3x}{4x^2 - 4} \)[/tex], we need to identify the values of [tex]\( x \)[/tex] for which the function is undefined. The function is undefined wherever its denominator is zero because division by zero is undefined.

Let's examine the denominator of the function:

[tex]\[ 4x^2 - 4 \][/tex]

First, we set the denominator equal to zero and solve for [tex]\( x \)[/tex]:

[tex]\[ 4x^2 - 4 = 0 \][/tex]

Next, we can factor out the common term:

[tex]\[ 4(x^2 - 1) = 0 \][/tex]

We recognize that [tex]\( x^2 - 1 \)[/tex] is a difference of squares:

[tex]\[ 4(x - 1)(x + 1) = 0 \][/tex]

Setting each factor equal to zero gives us:

[tex]\[ x - 1 = 0 \][/tex]
[tex]\[ x = 1 \][/tex]

[tex]\[ x + 1 = 0 \][/tex]
[tex]\[ x = -1 \][/tex]

So, the function [tex]\( f(x) = \frac{3x}{4x^2 - 4} \)[/tex] is undefined at [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex].

This means the domain of the function is all real numbers except [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex].

Therefore, the correct statement describing the domain of the function [tex]\( f(x) \)[/tex] is:

All real numbers except [tex]\( x = -1 \)[/tex] and [tex]\( x = 1 \)[/tex].

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