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Math  /  Algebra

QuestionExpress the function h(x)=1x+9h(x)=\frac{1}{x+9} in the form f∘gf \circ g. If g(x)=(x+9)g(x)=(x+9), find the function f(x)f(x). Your answer is f(x)=f(x)= □\square

Studdy Solution

STEP 1

1. We are given the function h(x)=1x+9 h(x) = \frac{1}{x+9} .
2. We need to express h(x) h(x) in the form f∘g f \circ g .
3. We are given g(x)=x+9 g(x) = x + 9 .
4. We need to find the function f(x) f(x) .

STEP 2

1. Understand the composition of functions.
2. Identify the inner function g(x) g(x) .
3. Determine the outer function f(x) f(x) such that f(g(x))=h(x) f(g(x)) = h(x) .
4. Express f(x) f(x) explicitly.

STEP 3

Understand that the composition of functions f∘g f \circ g means f(g(x)) f(g(x)) .

STEP 4

Identify the inner function g(x) g(x) .
Given: g(x)=x+9 g(x) = x + 9

STEP 5

Determine the outer function f(x) f(x) such that f(g(x))=h(x) f(g(x)) = h(x) .
We know: h(x)=1x+9 h(x) = \frac{1}{x+9}
Substitute g(x) g(x) into f f : f(g(x))=f(x+9)=1x+9 f(g(x)) = f(x+9) = \frac{1}{x+9}

STEP 6

From the equation f(x+9)=1x+9 f(x+9) = \frac{1}{x+9} , we can see that f(u)=1u f(u) = \frac{1}{u} where u=x+9 u = x+9 .

STEP 7

Express f(x) f(x) explicitly:
f(x)=1x f(x) = \frac{1}{x}
The function f(x) f(x) is:
f(x)=1x f(x) = \frac{1}{x}

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