# Given f(x) = -3x^2 - 8 and h(x) = 6x + 2 how do you find f(h(x))?

Mar 13, 2016

This means to plug function h in the place of x into f.

#### Explanation:

$f \left(h \left(x\right)\right) = - 3 {\left(6 x + 2\right)}^{2} - 8$

$f \left(h \left(x\right)\right) = - 3 \left(36 {x}^{2} + 24 x + 4\right) - 8$

$f \left(h \left(x\right)\right) = - 108 {x}^{2} - 72 x - 12 - 8$

$f \left(h \left(x\right)\right) = - 108 {x}^{2} - 72 x - 20$

Note that $f \left(h \left(x\right)\right) = \left(f \circ g\right) \left(x\right)$, they're just different notation forms.

When a number replaces the x in parentheses you must plug that number in for x in the inner function, find the result of that calculation, and then plug that in for x in the outer function. When there are three or more functions, make sure to work from the inside out, in the functions' respective order.

Practice exercises:

Assuming that $f \left(x\right) = - 2 x + 5 , g \left(x\right) = 2 {x}^{2} + 3 x - 2 \mathmr{and} h \left(x\right) = \sqrt{4 x - 3}$, find the following compositions.

a) $h \left(f \left(g \left(x\right)\right)\right)$

b) $g \left(h \left(f \left(x\right)\right)\right)$

c). $f \left(h \left(f \left(h \left(x\right)\right)\right)\right)$

d). $g \left(f \left(h \left(7\right)\right)\right)$