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Math  /  Algebra

QuestionLet f(x)=3x2f(x) = \frac{3}{x^2} and g(x)=1xg(x) = \frac{1}{x}.
9. Find f∘gf \circ g (f∘g)=f(1x)=3(1x)2(f \circ g) = f\left(\frac{1}{x}\right) = \frac{3}{(\frac{1}{x})^2} 3x23x^2 (f∘g)(x)=3x2(f \circ g)(x) = 3x^2
11. Find g∘fg \circ f

Studdy Solution

STEP 1

1. We are given two functions f(x)=3x2 f(x) = \frac{3}{x^2} and g(x)=1x g(x) = \frac{1}{x} .
2. We need to find the composition of these functions in the order specified, which means substituting one function into the other.

STEP 2

1. Determine the composition g∘f g \circ f .
2. Simplify the resulting expression.

STEP 3

To find g∘f g \circ f , substitute f(x) f(x) into g(x) g(x) . This means replacing x x in g(x)=1x g(x) = \frac{1}{x} with f(x)=3x2 f(x) = \frac{3}{x^2} .
g(f(x))=g(3x2)=13x2 g(f(x)) = g\left(\frac{3}{x^2}\right) = \frac{1}{\frac{3}{x^2}}

STEP 4

Simplify the expression 13x2 \frac{1}{\frac{3}{x^2}} .
13x2=x23 \frac{1}{\frac{3}{x^2}} = \frac{x^2}{3}
Thus, the composition (g∘f)(x) (g \circ f)(x) is:
(g∘f)(x)=x23 (g \circ f)(x) = \frac{x^2}{3}
The equivalent expression for g∘f g \circ f is:
x23 \boxed{\frac{x^2}{3}}

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