Math  /  Algebra

QuestionIf f(x)=x+1f(x)=x+1 and g(x)=x1g(x)=x-1, (a) f(g(x))=f(g(x))= \square (b) g(f(x))=g(f(x))= \square (c) Thus g(x)g(x) is called an \square function of f(x)f(x) Question Help: Video

Studdy Solution

STEP 1

1. The functions f(x)=x+1f(x) = x + 1 and g(x)=x1g(x) = x - 1 are defined for all real numbers xx.
2. We need to find the compositions f(g(x))f(g(x)) and g(f(x))g(f(x)).
3. We aim to identify the relationship between g(x)g(x) and f(x)f(x) based on their compositions.

STEP 2

1. Find the composition f(g(x))f(g(x)).
2. Find the composition g(f(x))g(f(x)).
3. Determine the relationship between g(x)g(x) and f(x)f(x) based on the compositions.

STEP 3

To find f(g(x))f(g(x)), substitute g(x)=x1g(x) = x - 1 into f(x)f(x).
f(g(x))=f(x1) f(g(x)) = f(x - 1)

STEP 4

Evaluate f(x1)f(x - 1) using the definition of f(x)f(x).
f(x1)=(x1)+1 f(x - 1) = (x - 1) + 1

STEP 5

Simplify the expression.
f(x1)=x f(x - 1) = x

STEP 6

To find g(f(x))g(f(x)), substitute f(x)=x+1f(x) = x + 1 into g(x)g(x).
g(f(x))=g(x+1) g(f(x)) = g(x + 1)

STEP 7

Evaluate g(x+1)g(x + 1) using the definition of g(x)g(x).
g(x+1)=(x+1)1 g(x + 1) = (x + 1) - 1

STEP 8

Simplify the expression.
g(x+1)=x g(x + 1) = x

STEP 9

Compare the results of f(g(x))f(g(x)) and g(f(x))g(f(x)).
f(g(x))=xandg(f(x))=x f(g(x)) = x \quad \text{and} \quad g(f(x)) = x

STEP 10

Since both compositions f(g(x))f(g(x)) and g(f(x))g(f(x)) return the original input xx, g(x)g(x) is an inverse function of f(x)f(x).
Thus, g(x)g(x) is called an inverse function of f(x)f(x).
Solution: (a) f(g(x))=xf(g(x)) = x (b) g(f(x))=xg(f(x)) = x (c) Thus g(x)g(x) is called an inverse function of f(x)f(x).

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