Find the domain of the composite functions [tex]$f \circ g$[/tex] and [tex]$g \circ f$[/tex].
[tex]$f(x)=\sqrt{4-x} ; \quad g(x)=\frac{1}{x-8}$[/tex]
Answer (2)
Find the domain of f ( x ) : x ≤ 4 , so the domain is ( − ∞ , 4 ] .
Find the domain of g ( x ) : x = 8 , so the domain is ( − ∞ , 8 ) ∪ ( 8 , ∞ ) .
Find the domain of f ∘ g ( x ) = f ( g ( x )) = 4 − x − 8 1 : ( − ∞ , 8 ) ∪ [ 4 33 , ∞ ) .
Find the domain of g ∘ f ( x ) = g ( f ( x )) = 4 − x − 8 1 : ( − ∞ , − 60 ) ∪ ( − 60 , 4 ] .
The domain of f ∘ g is ( − ∞ , 8 ) ∪ [ 4 33 , ∞ ) and the domain of g ∘ f is ( − ∞ , − 60 ) ∪ ( − 60 , 4 ] .
Explanation
Problem Setup We are given two functions, $f(x) =
\sqrt{4-x} an d g(x) = \frac{1}{x-8} , an d w e n ee d t o f in d t h e d o main so f t h eco m p os i t e f u n c t i o n s f \circ g an d g \circ f$.
Domain of f(x) First, let's find the domain of f ( x ) . Since we have a square root, the expression inside the square root must be non-negative: 4 − x ≥ 0 . Solving for x , we get x ≤ 4 . So, the domain of f ( x ) is ( − ∞ , 4 ] .
Domain of g(x) Next, let's find the domain of g ( x ) . Since we have a fraction, the denominator cannot be zero: x − 8 = 0 . Solving for x , we get x = 8 . So, the domain of g ( x ) is ( − ∞ , 8 ) ∪ ( 8 , ∞ ) .
Finding f(g(x)) Now, let's find the composite function f ∘ g ( x ) = f ( g ( x )) . We have f ( g ( x )) = f ( x − 8 1 ) = 4 − x − 8 1 . To find the domain of f ∘ g , we need to consider two conditions: x = 8 (because g ( x ) is defined only when x = 8 ) and 4 − x − 8 1 ≥ 0 (because the expression inside the square root must be non-negative).
Solving the Inequality Let's solve the inequality 4 − x − 8 1 ≥ 0 . This is equivalent to x − 8 4 ( x − 8 ) − 1 ≥ 0 , which simplifies to x − 8 4 x − 32 − 1 ≥ 0 , or x − 8 4 x − 33 ≥ 0 . The critical points are x = 8 and x = 4 33 = 8.25 . We analyze the sign of the expression x − 8 4 x − 33 in the intervals ( − ∞ , 8 ) , ( 8 , 4 33 ) , and ( 4 33 , ∞ ) .
Domain of f(g(x))
If x < 8 , then 4 x − 33 < 0 and x − 8 < 0 , so 0"> x − 8 4 x − 33 > 0 .
If 8 < x < 4 33 , then 4 x − 33 < 0 and 0"> x − 8 > 0 , so x − 8 4 x − 33 < 0 .
If \frac{33}{4}"> x > 4 33 , then 0"> 4 x − 33 > 0 and 0"> x − 8 > 0 , so 0"> x − 8 4 x − 33 > 0 .
If x = 4 33 , then x − 8 4 x − 33 = 0 .
Therefore, the solution to x − 8 4 x − 33 ≥ 0 is x < 8 or x ≥ 4 33 . Combining this with the condition x = 8 , the domain of f ∘ g is ( − ∞ , 8 ) ∪ [ 4 33 , ∞ ) .
Finding g(f(x)) Now, let's find the composite function g ∘ f ( x ) = g ( f ( x )) . We have g ( f ( x )) = g ( 4 − x ) = 4 − x − 8 1 . To find the domain of g ∘ f , we need to consider two conditions: 4 − x ≥ 0 (because of the square root) and 4 − x − 8 = 0 (because the denominator cannot be zero).
Domain of g(f(x)) The first condition, 4 − x ≥ 0 , implies x ≤ 4 . The second condition, 4 − x − 8 = 0 , implies 4 − x = 8 . Squaring both sides, we get 4 − x = 64 , which means x = − 60 . Therefore, the domain of g ∘ f is x ≤ 4 and x = − 60 . In interval notation, this is ( − ∞ , − 60 ) ∪ ( − 60 , 4 ] .
Final Answer In summary, the domain of f ∘ g is ( − ∞ , 8 ) ∪ [ 4 33 , ∞ ) , and the domain of g ∘ f is ( − ∞ , − 60 ) ∪ ( − 60 , 4 ] .
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 these composite functions ensures that the calculations are valid and meaningful.
The domain of f ∘ g is ( − ∞ , 8 ) ∪ [ 4 33 , ∞ ) , and the domain of g ∘ f is ( − ∞ , − 60 ) ∪ ( − 60 , 4 ] .
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