For [tex]f(x)=2 x+1[/tex] and [tex]g(x)=x^2-7[/tex], find [tex]\left(\frac{g}{f}\right)(x)[/tex].
A. [tex]\frac{2 x+1}{x^2-7}, x \neq \pm \sqrt{7}[/tex]
B. [tex]\frac{2 x+1}{x^2-7}[/tex]
C. [tex]\frac{x^2-7}{2 x+1}[/tex]
D. [tex]\frac{x^2-7}{2 x+1}, x=-\frac{1}{2}[/tex]
Answer (2)
Divide g ( x ) by f ( x ) to find f ( x ) g ( x ) = 2 x + 1 x 2 − 7 .
Set the denominator 2 x + 1 equal to zero and solve for x to find the restriction on the domain.
Solve 2 x + 1 = 0 to get x = − 2 1 .
State the final answer with the domain restriction: 2 x + 1 x 2 − 7 , x = − 2 1 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 2 x + 1 and g ( x ) = x 2 − 7 . Our goal is to find the expression for ( f g ) ( x ) , which means we need to divide g ( x ) by f ( x ) . We also need to identify any values of x that would make the denominator equal to zero, as division by zero is undefined.
Finding the Quotient First, let's find the expression for ( f g ) ( x ) :
( f g ) ( x ) = f ( x ) g ( x ) = 2 x + 1 x 2 − 7 This gives us the expression for the quotient of the two functions.
Finding the Restriction on the Domain Next, we need to find the values of x for which the denominator 2 x + 1 is equal to zero. This will tell us the values of x that are not allowed in the domain of the function ( f g ) ( x ) .
2 x + 1 = 0 Subtracting 1 from both sides, we get: 2 x = − 1 Dividing both sides by 2, we find: x = − 2 1
Stating the Domain Restriction Therefore, the function ( f g ) ( x ) = 2 x + 1 x 2 − 7 is defined for all real numbers x except x = − 2 1 . This is because when x = − 2 1 , the denominator 2 x + 1 becomes zero, making the function undefined.
Final Answer Now we can write the final answer, which includes the expression for ( f g ) ( x ) and the restriction on the domain: ( f g ) ( x ) = 2 x + 1 x 2 − 7 , x = − 2 1
Examples
Understanding the quotient of functions and their domains is crucial in many real-world applications. For instance, consider a scenario where g ( x ) represents the total cost of producing x items, and f ( x ) represents the number of workers available. Then, ( f g ) ( x ) would represent the average cost per worker for producing x items. Knowing the domain restrictions ensures that we don't divide by zero, which would be meaningless in this context. This concept is also vital in fields like physics, engineering, and economics, where functions often represent real-world quantities and their relationships.
The function ( f g ) ( x ) can be expressed as 2 x + 1 x 2 − 7 , with the restriction that x = − 2 1 because the denominator becomes zero at this point.
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