If [tex]m(x)=\frac{x-1}{x-1}[/tex] and [tex]n(x)=x-3[/tex], which function has the same domain as [tex](m \circ n)(x)[/tex]?
A. [tex]h(x)=\frac{x+5}{11}[/tex]
B. [tex]h(x)=\frac{11}{x-1}[/tex]
C. [tex]h(x)=\frac{11}{x-4}[/tex]
D. [tex]h(x)=\frac{11}{x-3}[/tex]
Answer (1)
Find the composite function ( m ∘ n ) ( x ) = m ( n ( x )) .
Determine the domain of ( m ∘ n ) ( x ) , which is all real numbers except x = 4 .
Find the domains of the given functions h ( x ) .
Compare the domains and identify the function with the same domain as ( m ∘ n ) ( x ) . The answer is h ( x ) = x − 4 11 .
Explanation
Understanding the Problem We are given two functions, m ( x ) = x − 1 x − 1 and n ( x ) = x − 3 , and we want to find which of the given functions has the same domain as the composite function ( m ∘ n ) ( x ) = m ( n ( x )) .
Finding the Composite Function First, let's find the composite function ( m ∘ n ) ( x ) . We have m ( n ( x )) = m ( x − 3 ) = ( x − 3 ) − 1 ( x − 3 ) − 1 = x − 4 x − 4 . Note that m ( x ) = x − 1 x − 1 is defined for all x = 1 , so m ( x ) = 1 for x = 1 . Thus, m ( n ( x )) = 1 for n ( x ) = 1 , which means x − 3 = 1 , so x = 4 .
Determining the Domain of the Composite Function Now, let's find the domain of ( m ∘ n ) ( x ) . Since m ( n ( x )) = x − 4 x − 4 , the function is undefined when the denominator is zero, i.e., x − 4 = 0 , which means x = 4 . Therefore, the domain of ( m ∘ n ) ( x ) is all real numbers except x = 4 .
Finding the Domains of the Given Functions Next, we need to find the domains of the given functions h ( x ) and compare them with the domain of ( m ∘ n ) ( x ) .
h ( x ) = 11 x + 5 : This function is defined for all real numbers since the denominator is a constant and never zero. So, the domain is all real numbers.
h ( x ) = x − 1 11 : This function is undefined when x − 1 = 0 , which means x = 1 . So, the domain is all real numbers except x = 1 .
h ( x ) = x − 4 11 : This function is undefined when x − 4 = 0 , which means x = 4 . So, the domain is all real numbers except x = 4 .
h ( x ) = x − 3 11 : This function is undefined when x − 3 = 0 , which means x = 3 . So, the domain is all real numbers except x = 3 .
Comparing the Domains Comparing the domains, we see that the domain of ( m ∘ n ) ( x ) is all real numbers except x = 4 , which is the same as the domain of h ( x ) = x − 4 11 .
Final Answer Therefore, the function that has the same domain as ( m ∘ n ) ( x ) is h ( x ) = x − 4 11 .
Examples
In electrical engineering, you might have a circuit where one component's output ( n ( x ) ) feeds into another component's input ( m ( x ) ). The composite function ( m ∘ n ) ( x ) describes the overall behavior of the combined components. Understanding the domain of this composite function is crucial to ensure the circuit operates correctly and avoids any undefined states or potential failures. For example, if n ( x ) represents the voltage output of a sensor and m ( x ) represents the response of an amplifier, the domain of ( m ∘ n ) ( x ) tells you the range of sensor outputs for which the amplifier will function properly.
