Which statement describes the domain of the function [tex]$f(x)=\frac{3 x}{4 x^2-4}$[/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 (2)
The domain of a rational function excludes values where the denominator is zero.
Set the denominator 4 x 2 − 4 equal to zero and solve for x .
Solving 4 x 2 − 4 = 0 gives x = ± 1 .
The domain is all real numbers except x = − 1 and x = 1 , so the answer is all real numbers except x = − 1 and x = 1 .
Explanation
Understanding the Domain of a Rational Function We are asked to find the domain of the function f ( x ) = f r a c 3 x 4 x 2 − 4 . The domain of a rational function consists of all real numbers except for the values of x that make the denominator equal to zero.
Setting the Denominator to Zero To find the values of x that make the denominator zero, we need to solve the equation 4 x 2 − 4 = 0 .
Solving for x We can solve this equation as follows:
4 x 2 − 4 = 0
Divide both sides by 4:
x 2 − 1 = 0
Add 1 to both sides:
x 2 = 1
Take the square root of both sides:
x = p m 1
So, x = 1 or x = − 1 .
Determining the Domain Therefore, the domain of the function f ( x ) = f r a c 3 x 4 x 2 − 4 is all real numbers except x = − 1 and x = 1 .
Examples
Consider a scenario where you are designing a bridge. The function f ( x ) = f r a c 3 x 4 x 2 − 4 might represent the stress on a certain part of the bridge, where x is a variable related to the load. It is crucial to know the values of x for which the function is undefined (in this case, x = − 1 and x = 1 ), as these values could represent critical points where the bridge could fail. Understanding the domain helps engineers ensure the bridge's safety by avoiding these critical values.
The domain of the function f ( x ) = 4 x 2 − 4 3 x excludes the values that make the denominator zero, which are x = − 1 and x = 1 . Thus, the domain consists of all real numbers except these two values. The correct answer is D: all real numbers except x = − 1 and x = 1 .
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