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Math  /  Algebra

QuestionSuppose f(x)=x−1xf(x)=\frac{x-1}{x} and g(x)=11−xg(x)=\frac{1}{1-x}
Then (f∘g)(x)=(f \circ g)(x)= □\square , and (g∘f)(x)=(g \circ f)(x)= □\square . Remarkable!

Studdy Solution

STEP 1

1. The composition of functions (f∘g)(x)(f \circ g)(x) and (g∘f)(x)(g \circ f)(x) requires substituting one function into the other.
2. The function f(x)=x−1xf(x) = \frac{x-1}{x} and g(x)=11−xg(x) = \frac{1}{1-x} must be substituted and simplified.
3. The domain restrictions of f(x)f(x) and g(x)g(x) need to be considered to ensure valid operations.

STEP 2

1. Compute (f∘g)(x)(f \circ g)(x) by substituting g(x)g(x) into f(x)f(x).
2. Simplify the expression for (f∘g)(x)(f \circ g)(x).
3. Compute (g∘f)(x)(g \circ f)(x) by substituting f(x)f(x) into g(x)g(x).
4. Simplify the expression for (g∘f)(x)(g \circ f)(x).

STEP 3

Substitute g(x)g(x) into f(x)f(x) to find (f∘g)(x)(f \circ g)(x).
(f∘g)(x)=f(g(x))=f(11−x) (f \circ g)(x) = f(g(x)) = f\left(\frac{1}{1-x}\right)

STEP 4

Using the definition of f(x)f(x), substitute 11−x\frac{1}{1-x} for xx in f(x)f(x).
f(11−x)=11−x−111−x f\left(\frac{1}{1-x}\right) = \frac{\frac{1}{1-x} - 1}{\frac{1}{1-x}}

STEP 5

Simplify the numerator of the fraction.
11−x−111−x=1−(1−x)1−x11−x=x1−x11−x \frac{\frac{1}{1-x} - 1}{\frac{1}{1-x}} = \frac{\frac{1-(1-x)}{1-x}}{\frac{1}{1-x}} = \frac{\frac{x}{1-x}}{\frac{1}{1-x}}

STEP 6

Simplify the overall fraction.
x1−x11−x=x \frac{\frac{x}{1-x}}{\frac{1}{1-x}} = x

STEP 7

Substitute f(x)f(x) into g(x)g(x) to find (g∘f)(x)(g \circ f)(x).
(g∘f)(x)=g(f(x))=g(x−1x) (g \circ f)(x) = g(f(x)) = g\left(\frac{x-1}{x}\right)

STEP 8

Using the definition of g(x)g(x), substitute x−1x\frac{x-1}{x} for xx in g(x)g(x).
g(x−1x)=11−x−1x g\left(\frac{x-1}{x}\right) = \frac{1}{1 - \frac{x-1}{x}}

STEP 9

Simplify the denominator of the fraction.
11−x−1x=1x−(x−1)x=1x−x+1x=11x \frac{1}{1 - \frac{x-1}{x}} = \frac{1}{\frac{x - (x-1)}{x}} = \frac{1}{\frac{x - x + 1}{x}} = \frac{1}{\frac{1}{x}}

STEP 10

Simplify the overall fraction.
11x=x \frac{1}{\frac{1}{x}} = x
Solution: (f∘g)(x)=x (f \circ g)(x) = x (g∘f)(x)=x (g \circ f)(x) = x

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