Let [tex]$f(x)=\sqrt{x-3}$[/tex] and [tex]$g(x)=\frac{1}{x}$[/tex]. Find the domain of the composite function [tex]$(g \circ f)(x)=g(f(x))$[/tex].
Answer (2)
The domain of the composite function ( g ∘ f ) ( x ) = g ( f ( x )) is all real numbers greater than 3, represented in interval notation as ( 3 , ∞ ) .
;
Find the composite function: g ( f ( x )) = x − 3 1 .
Determine the restrictions: x − 3 ≥ 0 and x − 3 = 0 .
Solve the inequalities: x ≥ 3 and x = 3 .
Combine the restrictions to find the domain: 3"> x > 3 , which in interval notation is ( 3 , ∞ ) .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x − 3 and g ( x ) = x 1 , and we want to find the domain of the composite function ( g ∘ f ) ( x ) = g ( f ( x )) . This means we need to find all possible values of x for which the composite function is defined.
Finding the Composite Function First, let's find the expression for the composite function g ( f ( x )) . We substitute f ( x ) into g ( x ) : g ( f ( x )) = g ( x − 3 ) = x − 3 1 .
Identifying Restrictions Now, we need to determine the domain of g ( f ( x )) = x − 3 1 . There are two restrictions we need to consider:
The expression inside the square root must be non-negative: x − 3 ≥ 0 , which means x ≥ 3 .
The denominator cannot be zero: x − 3 = 0 , which means x − 3 = 0 , so x = 3 .
Combining Restrictions Combining these two restrictions, we have x ≥ 3 and x = 3 . This means that x must be strictly greater than 3. Therefore, the domain of the composite function g ( f ( x )) is all real numbers x such that 3"> x > 3 .
Final Answer In interval notation, the domain of g ( f ( x )) is ( 3 , ∞ ) .
Examples
Composite functions are used in many real-world applications. For example, in manufacturing, if f ( x ) represents the number of products produced by x employees and g ( y ) represents the profit from selling y products, then g ( f ( x )) represents the profit from the products produced by x employees. Understanding the domain of composite functions helps businesses determine the number of employees needed to achieve a certain profit level.
