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QuestionFind the composite function for f(x)=2x−6f(x)=\frac{2}{x-6} and g(x)=32xg(x)=\frac{3}{2 x}. Compute (f∘g)(x)(f \circ g)(x).

Studdy Solution

STEP 1

Assumptions1. The given functions are f(x)=x−6f(x)=\frac{}{x-6} and g(x)=3xg(x)=\frac{3}{x}. . We need to find the composite function (f∘g)(x)(f \circ g)(x).

STEP 2

The composite function (f∘g)(x)(f \circ g)(x) is defined as f(g(x))f(g(x)). This means we substitute g(x)g(x) into the function f(x)f(x).

STEP 3

Substitute g(x)g(x) into f(x)f(x).
(f∘g)(x)=f(g(x))=f(32x)(f \circ g)(x) = f(g(x)) = f\left(\frac{3}{2x}\right)

STEP 4

Now, we substitute 32x\frac{3}{2x} into f(x)f(x) in place of xx.
(f∘g)(x)=232x−6(f \circ g)(x) = \frac{2}{\frac{3}{2x}-6}

STEP 5

To simplify the expression, we can multiply the denominator by 2x2x to get rid of the fraction.
(f∘g)(x)=22x⋅(32x−)(f \circ g)(x) = \frac{2}{2x \cdot \left(\frac{3}{2x}-\right)}

STEP 6

implify the expression.
(f∘g)(x)=23−12x(f \circ g)(x) = \frac{2}{3-12x}

STEP 7

Now, we can simplify the fraction by dividing the numerator and the denominator by2.
(f∘g)(x)=132−6x(f \circ g)(x) = \frac{1}{\frac{3}{2}-6x}

STEP 8

Finally, we simplify the expression to get the composite function (f∘g)(x)(f \circ g)(x).
(f∘g)(x)=13−12x2=23−12x(f \circ g)(x) = \frac{1}{\frac{3-12x}{2}} = \frac{2}{3-12x}So, the composite function (f∘g)(x)(f \circ g)(x) is 23−12x\frac{2}{3-12x}.

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