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=3\} [/tex]
Answer (1)
Find the composite function f ( g ( x )) by substituting g ( x ) into f ( x ) , resulting in f ( g ( x )) = 5 x − 4 5 x − 7 .
Determine the values of x for which the denominator of f ( g ( x )) is zero by solving 5 x − 4 = 0 .
Find that x = 5 4 makes the denominator zero, which must be excluded from the domain.
The domain of ( f v er t c i rc g ) ( x ) is all real numbers except x = 5 4 , represented as { 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 v er t c i rc g ) ( x ) = f ( g ( x )) . This means we need to find all possible values of x for which the function f ( g ( x )) is defined.
Finding the Composite Function First, let's find the expression for f ( g ( x )) . We substitute g ( x ) into f ( x ) : f ( g ( x )) = f ( 5 x − 4 ) = 5 x − 4 ( 5 x − 4 ) − 3 = 5 x − 4 5 x − 7 .
Determining the Domain Now, we need to determine the domain of f ( g ( x )) = 5 x − 4 5 x − 7 . The domain is all real numbers except for any values of x 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 To find the values of x that make the denominator zero, we solve the equation 5 x − 4 = 0 : 5 x − 4 = 0 ⟹ 5 x = 4 ⟹ x = 5 4 .
Final Answer Therefore, the domain of ( f v er t c i rc g ) ( x ) is all real numbers except x = 5 4 . In set notation, this is { x ∣ x = 5 4 } .
Examples
Imagine you are designing a machine where one part's output feeds directly into another. If g ( x ) represents the input to the first part and f ( x ) represents how the second part processes the output, then f ( g ( x )) describes the entire process. The domain of f ( g ( x )) tells you what inputs x will not cause any errors or breakdowns in the system. For example, if x = 5 4 causes a division by zero in our function, it means that input would break the machine, so we must exclude it.
