Question

For the functions f(x)=5x−3 and g(x)=6x2−3, find (f∘g)(x) and (g∘f)(x)

Answer 1

Answers:

$(f \circ g)(x) = 30x^2-18\\\\(g \circ f)(x) = 150x^2-180x+51$

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Work Shown:

$f(x) = 5x-3\\\\f(g(x)) = 5(g(x))-3\\\\(f \circ g)(x) = 5(6x^2-3)-3\\\\(f \circ g)(x) = 30x^2-15-3\\\\(f \circ g)(x) = 30x^2-18$

Those steps show we start with the outer function f(x) = 5x-3. Then we replace every x with g(x) in the second step. Afterward, we plug in g(x) = 6x^2-3 and we simplify.

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We'll follow similar steps for (g∘f)(x)

But this time we started with g(x), plug in f(x) for every x, and then simplified.

$g(x) = 6x^2-3\\\\g(f(x)) = 6(f(x))^2-3\\\\(g \circ f)(x) = 6(5x-3)^2-3\\\\(g \circ f)(x) = 6(25x^2-30x+9)-3\\\\(g \circ f)(x) = 150x^2-180x+54-3\\\\(g \circ f)(x) = 150x^2-180x+51$