For [tex]f(x)=4 x+1[/tex] and [tex]g(x)=x^2-5[/tex], find [tex]\left(\frac{f}{g}\right)(x)[/tex]
A. [tex]\frac{4 x+1}{x^2-5}[/tex]
B. [tex]\frac{x^2-5}{4 x+1}[/tex]
C. [tex]\frac{x^2-5}{4 x+1}, x \neq-\frac{1}{4}[/tex]
D. [tex]\frac{4 x+1}{x^2-5}, x \neq \pm \sqrt{5}[/tex]
Answer (2)
Divide the functions: ( g f ) ( x ) = x 2 − 5 4 x + 1 .
Find the values where the denominator is zero: x 2 − 5 = 0 .
Solve for x : x = ± 5 .
State the final answer with restrictions: x 2 − 5 4 x + 1 , x = ± 5 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 4 x + 1 and g ( x ) = x 2 − 5 . We need to find ( g f ) ( x ) , which means we need to divide f ( x ) by g ( x ) .
Dividing the Functions To find ( g f ) ( x ) , we divide f ( x ) by g ( x ) : ( g f ) ( x ) = g ( x ) f ( x ) = x 2 − 5 4 x + 1
Finding Restrictions on the Domain Now, we need to find the restrictions on the domain of the resulting function. The domain is restricted when the denominator is equal to zero. So, we need to find the values of x for which x 2 − 5 = 0 .
Solving for the Restricted Values To solve x 2 − 5 = 0 , we can add 5 to both sides: x 2 = 5 Taking the square root of both sides, we get: x = ± 5 So, the values x = 5 and x = − 5 make the denominator zero, and these values must be excluded from the domain.
Final Answer Therefore, the expression for ( g f ) ( x ) is x 2 − 5 4 x + 1 , and the restrictions on the domain are x = ± 5 . This means the correct answer is D.
Examples
Understanding function division and domain restrictions is crucial in many real-world applications. For example, consider a scenario where the function f ( x ) represents the total cost of producing x items, and g ( x ) represents the number of items produced. Then, ( g f ) ( x ) would represent the average cost per item. It's important to identify any restrictions on x (the domain) to ensure that the average cost is defined and meaningful. For instance, if g ( x ) = x 2 − 5 , then x cannot be ± 5 , as this would lead to division by zero, making the average cost undefined.
The function ( g f ) ( x ) is calculated as x 2 − 5 4 x + 1 . The values x = ± 5 must be excluded from the domain due to the denominator being zero at these points. Therefore, the correct answer is option D.
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