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Math  /  Algebra

Question- For f(x)=9x−8f(x)=\frac{9}{x-8} and g(x)=1xg(x)=\frac{1}{x}, find the following composite functions and state the domain of each. (a) f∘gf \circ g (b) g∘fg \circ f (c) f∘ff \circ f (d) g∘g\mathrm{g} \circ \mathrm{g} (a) (f∘g)(x)=(f \circ g)(x)= □\square (Simplify your answer.)

Studdy Solution

STEP 1

1. We are given two functions f(x)=9x−8 f(x) = \frac{9}{x-8} and g(x)=1x g(x) = \frac{1}{x} .
2. We need to find the composite functions f∘g f \circ g , g∘f g \circ f , f∘f f \circ f , and g∘g g \circ g .
3. We need to simplify each composite function and determine its domain.

STEP 2

1. Find and simplify the composite function f∘g f \circ g .
2. Determine the domain of f∘g f \circ g .
3. Find and simplify the composite function g∘f g \circ f .
4. Determine the domain of g∘f g \circ f .
5. Find and simplify the composite function f∘f f \circ f .
6. Determine the domain of f∘f f \circ f .
7. Find and simplify the composite function g∘g g \circ g .
8. Determine the domain of g∘g g \circ g .

STEP 3

To find (f∘g)(x) (f \circ g)(x) , substitute g(x) g(x) into f(x) f(x) :
(f∘g)(x)=f(g(x))=f(1x) (f \circ g)(x) = f(g(x)) = f\left(\frac{1}{x}\right)
Substitute 1x \frac{1}{x} into f(x) f(x) :
f(1x)=91x−8 f\left(\frac{1}{x}\right) = \frac{9}{\frac{1}{x} - 8}
Simplify the expression:
=91−8xx=9x1−8x = \frac{9}{\frac{1 - 8x}{x}} = \frac{9x}{1 - 8x}
So, (f∘g)(x)=9x1−8x (f \circ g)(x) = \frac{9x}{1 - 8x} .

STEP 4

Determine the domain of f∘g f \circ g :
1. The expression g(x)=1x g(x) = \frac{1}{x} is undefined for x=0 x = 0 .
2. The expression f(1x)=91x−8 f\left(\frac{1}{x}\right) = \frac{9}{\frac{1}{x} - 8} is undefined when 1x−8=0 \frac{1}{x} - 8 = 0 .
Solve for x x :
1x=8⇒x=18 \frac{1}{x} = 8 \Rightarrow x = \frac{1}{8}
Thus, the domain of f∘g f \circ g is all real numbers except x=0 x = 0 and x=18 x = \frac{1}{8} .
The simplified expression for (f∘g)(x) (f \circ g)(x) is:
9x1−8x \boxed{\frac{9x}{1 - 8x}}
The domain of f∘g f \circ g is x≠0 x \neq 0 and x≠18 x \neq \frac{1}{8} .

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