QuestionFind the composite functions for: a) $f(x)=3x+2$ , b) $g(h(x))$ , c) $h(g(f(x)))$ .

Studdy Solution

STEP 1

Assumptions1. The function f(x) is defined as $f(x) =3x +$ . The function g(x) is defined as $g(x) = x^3$
3. The function h(x) is defined as $h(x) = \cos x$

STEP 2

For part a), we are asked to find $f(x)$ . Since $f(x)$ is already given, there is no calculation needed. $f(x) =x +2$

STEP 3

For part b), we are asked to find $g(h(x))$ . This means we substitute $h(x)$ into $g(x)$ . $g(h(x)) = (h(x))^3$

STEP 4

Now, we substitute the given function $h(x) = \cos x$ into the equation.
$g(h(x)) = (\cos x)^3$

STEP 5

For part c), we are asked to find $h(g(f(x)))$ . This means we substitute $f(x)$ into $g(x)$ , and then substitute that result into $h(x)$ .First, substitute $f(x)$ into $g(x)$ $g(f(x)) = (f(x))^3$

STEP 6

Substitute the given function $f(x) =3x +2$ into the equation.
$g(f(x)) = (3x +2)^3$

STEP 7

Now, substitute $g(f(x))$ into $h(x)$ $h(g(f(x))) = \cos(g(f(x)))$

STEP 8

Substitute the result from step6 into the equation.
$h(g(f(x))) = \cos((3x +2)^3)$ So, the composite functions area) $f(x) =3x +2$
b) $g(h(x)) = (\cos x)^3$
c) $h(g(f(x))) = \cos((3x +2)^3)$

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