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Math

Math Snap

PROBLEM

Check if f(x)=x+0f(x)=x+0 and g(x)=x+18g(x)=x+18 are inverses by finding (fg)(x)(f \circ g)(x) and (gf)(x)(g \circ f)(x).

STEP 1

Assumptions1. The function f(x)f(x) is defined as f(x)=x+0f(x)=x+0
. The function g(x)g(x) is defined as g(x)=x+18g(x)=x+18
3. We need to determine if f(x)f(x) and g(x)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 ff and gg are inverse functions, then the composition of ff and gg in both orders should return the original input. Mathematically, this means thatf(g(x))=xandg(f(x))=xf(g(x)) = x \quad \text{and} \quad g(f(x)) = x

STEP 3

Let's first compute the composition (fg)(x)(f \circ g)(x), which means we apply the function ff to the output of g(x)g(x).
(fg)(x)=f(g(x)) (f \circ g)(x) = f(g(x))

STEP 4

Substitute g(x)g(x) into the function f(x)f(x).
(fg)(x)=f(x+18) (f \circ g)(x) = f(x+18)

STEP 5

Substitute x+18x+18 into f(x)f(x).
(fg)(x)=(x+18)+0 (f \circ g)(x) = (x+18) +0

STEP 6

implify the expression.
(fg)(x)=x+18 (f \circ g)(x) = x+18

STEP 7

Now, let's compute the composition (gf)(x)(g \circ f)(x), which means we apply the function gg to the output of f(x)f(x).
(gf)(x)=g(f(x)) (g \circ f)(x) = g(f(x))

STEP 8

Substitute f(x)f(x) into the function g(x)g(x).
(gf)(x)=g(x+0) (g \circ f)(x) = g(x+0)

STEP 9

Substitute x+x+ into g(x)g(x).
(gf)(x)=(x+)+18 (g \circ f)(x) = (x+) +18

STEP 10

implify the expression.
(gf)(x)=x+18 (g \circ f)(x) = x+18

SOLUTION

Now, we compare the results of (fg)(x)(f \circ g)(x) and (gf)(x)(g \circ f)(x) with xx. Neither of them equals to xx. Therefore, f(x)f(x) and g(x)g(x) are not inverse functions.
Thus, g(x)g(x) is not the inverse function of f(x)f(x).

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