Find the domain of the composite functions [tex]$f \circ g$[/tex] and [tex]$g \circ f$[/tex].
[tex]$f(x)=\sqrt{5-x} ; \quad g(x)=\frac{1}{x-7}$[/tex]

What is the domain of the composite function [tex]$f \circ g$[/tex]?

Find the domain of f ( x ) : x ≤ 5 , so the domain is ( − ∞ , 5 ] .
Find the domain of g ( x ) : x e q 7 , so the domain is ( − ∞ , 7 ) c u p ( 7 , ∞ ) .
Find the domain of f ( g ( x )) : ( − ∞ , 7 ) ∪ [ 7.2 , ∞ ) .
Find the domain of g ( f ( x )) : ( − ∞ , − 44 ) ∪ ( − 44 , 5 ] .

The domain of f ( g ( x )) is ( − ∞ , 7 ) ∪ [ 7.2 , ∞ ) ​ . The domain of g ( f ( x )) is ( − ∞ , − 44 ) ∪ ( − 44 , 5 ] ​ .
Explanation

Understanding the Problem We are given two functions, f ( x ) = 5 − x ​ and g ( x ) = x − 7 1 ​ , and we need to find the domains of the composite functions f ∘ g and g ∘ f . This involves understanding the individual domains of f ( x ) and g ( x ) and how they affect the composite functions.

Domain of f(x) First, let's find the domain of f ( x ) . Since f ( x ) = 5 − x ​ , the expression inside the square root must be non-negative. Therefore, 5 − x ≥ 0 , which implies x ≤ 5 . So, the domain of f ( x ) is ( − ∞ , 5 ] .

Domain of g(x) Next, let's find the domain of g ( x ) . Since g ( x ) = x − 7 1 ​ , the denominator cannot be zero. Therefore, x − 7  = 0 , which means x  = 7 . So, the domain of g ( x ) is ( − ∞ , 7 ) ∪ ( 7 , ∞ ) .

Finding f(g(x)) Now, let's find the composite function f ( g ( x )) = f ( x − 7 1 ​ ) = 5 − x − 7 1 ​ ​ . For f ( g ( x )) to be defined, we need two conditions to be satisfied: 1) x must be in the domain of g ( x ) , so x  = 7 , and 2) the expression inside the square root must be non-negative, so 5 − x − 7 1 ​ ≥ 0 . This means x − 7 1 ​ ≤ 5 .

Domain of f(g(x)) Let's solve the inequality x − 7 1 ​ ≤ 5 . We consider two cases:


Case 1: 7"> x > 7 . In this case, 0"> x − 7 > 0 , so we can multiply both sides of the inequality by x − 7 without changing the direction of the inequality: 1 ≤ 5 ( x − 7 ) , which simplifies to 1 ≤ 5 x − 35 . Adding 35 to both sides gives 36 ≤ 5 x , so x ≥ 5 36 ​ = 7.2 . Thus, in this case, we have x ≥ 7.2 .
Case 2: x < 7 . In this case, x − 7 < 0 , so when we multiply both sides of the inequality by x − 7 , we must reverse the direction of the inequality: 1 ≥ 5 ( x − 7 ) , which simplifies to 1 ≥ 5 x − 35 . Adding 35 to both sides gives 36 ≥ 5 x , so x ≤ 5 36 ​ = 7.2 . Since we assumed x < 7 , we have x < 7 .
Combining these two cases, the domain of f ( g ( x )) is ( − ∞ , 7 ) ∪ [ 7.2 , ∞ ) .

Finding g(f(x)) Now, let's find the composite function g ( f ( x )) = g ( 5 − x ​ ) = 5 − x ​ − 7 1 ​ . For g ( f ( x )) to be defined, we need two conditions to be satisfied: 1) x must be in the domain of f ( x ) , so x ≤ 5 , and 2) the denominator cannot be zero, so 5 − x ​ − 7  = 0 . This means 5 − x ​  = 7 , so 5 − x  = 49 , which implies x  = 5 − 49 = − 44 .

Domain of g(f(x)) Therefore, the domain of g ( f ( x )) is ( − ∞ , − 44 ) ∪ ( − 44 , 5 ] .

Final Answer for f(g(x)) The domain of the composite function f ∘ g is ( − ∞ , 7 ) ∪ [ 7.2 , ∞ ) .

Final Answer for g(f(x)) The domain of the composite function g ∘ f is ( − ∞ , − 44 ) ∪ ( − 44 , 5 ] .


Examples
Composite functions are useful in many real-world applications. For example, consider a store that offers a discount of 10% on all items and then applies a sales tax of 5%. If f ( x ) = 0.9 x represents the price after the discount and g ( x ) = 1.05 x represents the price after the sales tax, then the composite function g ( f ( x )) represents the final price of an item after both the discount and the sales tax are applied. Understanding the domain of such composite functions ensures that the calculations are valid and meaningful.

Answered by GinnyAnswer | 2025-07-03

The domain of the composite function f ∘ g is ( − ∞ , 7 ) ∪ [ 7.2 , ∞ ) and the domain of g ∘ f is ( − ∞ , − 44 ) ∪ ( − 44 , 5 ] .
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Answered by Anonymous | 2025-07-04