If [tex]$m(x)=\frac{x+5}{x-1}$[/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 (2)
Find the composite function ( m ∘ n ) ( x ) by substituting n ( x ) into m ( x ) , resulting in ( m ∘ n ) ( x ) = x − 4 x + 2 .
Determine the domain of ( m ∘ n ) ( x ) by finding the values of x that make the denominator zero, which is x = 4 . Thus, the domain is all real numbers except x = 4 .
Find the domains of the given functions and compare them to the domain of ( m ∘ n ) ( x ) .
The function h ( x ) = x − 4 11 has the same domain as ( m ∘ n ) ( x ) .
h ( x ) = x − 4 11
Explanation
Finding the composite function First, we need to find the expression for ( m c i rc n ) ( x ) . This means we substitute n ( x ) into m ( x ) wherever we see x . Given $m(x) =
x − 1 x + 5
and n ( x ) = x − 3 , we have
( m c i rc n ) ( x ) = m ( n ( x )) = m ( x − 3 ) = ( x − 3 ) − 1 ( x − 3 ) + 5 = x − 4 x + 2 .
Determining the domain of the composite function Next, we determine the domain of ( m c i rc n ) ( x ) = x − 4 x + 2 . The domain of a rational function is all real numbers except for the values of x that make the denominator equal to zero. So, we need to find the values of x for which x − 4 = 0 . Solving for x , we get x = 4 . Therefore, the domain of ( m c i rc n ) ( x ) is all real numbers except x = 4 . In interval notation, this is $(-\infty, 4)
\cup (4, \infty)$.
Finding the domains of the given functions Now, we need to find which of the given functions has the same domain as ( m c i rc n ) ( x ) . Let's examine each option:
h ( x ) = 11 x + 5 : The denominator is a constant, 11, which is never zero. Thus, the domain of this function is all real numbers.
h ( x ) = x − 1 11 : The denominator is x − 1 , which is zero when x = 1 . Thus, the domain of this function is all real numbers except x = 1 .
h ( x ) = x − 4 11 : The denominator is x − 4 , which is zero when x = 4 . Thus, the domain of this function is all real numbers except x = 4 .
h ( x ) = x − 3 11 : The denominator is x − 3 , which is zero when x = 3 . Thus, the domain of this function is all real numbers except x = 3 .
Comparing the domains Comparing the domains, we see that the function h ( x ) = x − 4 11 has the same domain as ( m c i rc n ) ( x ) , which is all real numbers except x = 4 .
Final Answer Therefore, the function with the same domain as ( m c i rc n ) ( x ) is h ( x ) = x − 4 11 .
Examples
Understanding the domain of composite functions is crucial in many real-world applications. For example, consider a scenario where a company's profit, m ( x ) , depends on the number of products sold, x , and the number of products sold, n ( x ) , depends on the amount spent on advertising. The composite function ( m ∘ n ) ( x ) represents the company's profit as a function of the amount spent on advertising. Knowing the domain of this composite function helps the company determine the range of advertising spending that will result in a valid and meaningful profit. If the domain excludes certain values, it indicates constraints on advertising spending to avoid negative profits or other undesirable outcomes. This ensures that the company makes informed decisions about its advertising budget to maximize profitability.
The function with the same domain as ( m ∘ n ) ( x ) is h ( x ) = x − 4 11 , which is defined for all real numbers except x = 4 . This is because the composite function's denominator is zero at x = 4 . Therefore, the correct answer is option C.
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