PROBLEM
Check if f(x)=x+0 and g(x)=x+18 are inverses by finding (f∘g)(x) and (g∘f)(x).
STEP 1
Assumptions1. The function f(x) is defined as f(x)=x+0
. The function g(x) is defined as g(x)=x+18
3. We need to determine if f(x) and g(x) are inverse functions by using function composition.
STEP 2
First, we need to understand what it means for two functions to be inverses of each other. If f and g are inverse functions, then the composition of f and g in both orders should return the original input. Mathematically, this means thatf(g(x))=xandg(f(x))=x
STEP 3
Let's first compute the composition (f∘g)(x), which means we apply the function f to the output of g(x).
(f∘g)(x)=f(g(x))
STEP 4
Substitute g(x) into the function f(x).
(f∘g)(x)=f(x+18)
STEP 5
Substitute x+18 into f(x).
(f∘g)(x)=(x+18)+0
STEP 6
implify the expression.
(f∘g)(x)=x+18
STEP 7
Now, let's compute the composition (g∘f)(x), which means we apply the function g to the output of f(x).
(g∘f)(x)=g(f(x))
STEP 8
Substitute f(x) into the function g(x).
(g∘f)(x)=g(x+0)
STEP 9
Substitute x+ into g(x).
(g∘f)(x)=(x+)+18
STEP 10
implify the expression.
(g∘f)(x)=x+18
SOLUTION
Now, we compare the results of (f∘g)(x) and (g∘f)(x) with x. Neither of them equals to x. Therefore, f(x) and g(x) are not inverse functions.
Thus, g(x) is not the inverse function of f(x).
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