If [tex]f(x)=\frac{x-3}{x}[/tex] and [tex]g(x)=5 x-4[/tex], what is the domain of [tex](f \circ g)(x)[/tex] ?
A. [tex] \{x \mid x \neq 0\} [/tex]
B. [tex]\left\{x \left\lvert\, x \neq \frac{1}{3}\right.\right\}[/tex]
C. [tex]\left\{x \left\lvert\, x \neq \frac{4}{5}\right.\right\}[/tex]
D. [tex] \{x \mid x \neq 3\} [/tex]
Answer (2)
Find the composite function: ( f ∘ g ) ( x ) = f ( g ( x )) = 5 x − 4 5 x − 7 .
Determine the values of x that make the denominator zero: 5 x − 4 = 0 .
Solve for x : x = 5 4 .
The domain of ( f ∘ g ) ( x ) is all real numbers except x = 5 4 , which is { x ∣ x = 5 4 } .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x x − 3 and g ( x ) = 5 x − 4 . We want to find the domain of the composite function ( f ∘ g ) ( x ) = f ( g ( x )) . This means we first evaluate g ( x ) and then plug the result into f ( x ) .
Finding the Composite Function First, let's find the expression for ( f ∘ g ) ( x ) . We have
( f ∘ g ) ( x ) = f ( g ( x )) = f ( 5 x − 4 ) = 5 x − 4 ( 5 x − 4 ) − 3 = 5 x − 4 5 x − 7 .
Finding the Domain Now we need to find the domain of this composite function. The domain of a rational function is all real numbers except for the values that make the denominator equal to zero. So, we need to find the values of x for which 5 x − 4 = 0 .
Solving for x We set the denominator equal to zero and solve for x :
5 x − 4 = 0
5 x = 4 x = 5 4
Final Answer Therefore, the domain of ( f ∘ g ) ( x ) is all real numbers except x = 5 4 . In set notation, this is written as { x ∣ x = 5 4 } .
Examples
Composite functions are useful in many real-world scenarios. For example, consider a store that offers a discount of 10% on all items and then applies a sales tax of 5%. If x is the original price of an item, the discounted price is g ( x ) = 0.9 x , and the price after tax is f ( x ) = 1.05 x . The final price after both the discount and tax are applied is ( f ∘ g ) ( x ) = f ( g ( x )) = 1.05 ( 0.9 x ) = 0.945 x . Understanding the domain of composite functions helps in ensuring that the inputs are valid and the outputs are meaningful in practical applications.
The domain of the composite function ( f ∘ g ) ( x ) is all real numbers except x = 5 4 . This is represented as { x ∣ x = 5 4 } . Therefore, the correct answer is option C.
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