# Given f(x) = x^2 + x + 1 h(x) = 3x + 2, evaluate the composite function.

Given
$f\left(x\right)={x}^{2}+x+1$
h(x) = 3x + 2,
evaluate the composite function.
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Nicole Conner
By using the definition of a composite function,
$\left(f\circ g\right)\left(x\right)=f\left(h\left(x\right)\right)$
$\left(f\circ g\right)\left(x\right)=f\left(h\left(x\right)\right)$
=f(3x+2)
To find value of f(3x+2) replace x by 3x+2 in function f.
$f\left(3x+2\right)={\left(3x+2\right)}^{2}+\left(3x+2\right)+1$
$=9{x}^{2}+12x+4+3x+2+1$
$=9{x}^{2}+15x+7$
Thus, $\left(f\circ h\right)\left(x\right)=9{x}^{2}+15x+7$
Again by using the definition of a composite function,
$\left(h\circ f\right)\left(x\right)=h\left(f\left(x\right)\right)$
$\left(h\circ f\right)\left(x\right)=h\left(f\left(x\right)\right)$
$=h\left({x}^{2}+x+1\right)$
To find $h\left({x}^{2}+x+1\right)$ replace $x\in h\left(x\right)by{x}^{2}+x+1$
$h\left({x}^{2}+x+1\right)=3\left({x}^{2}+x+1\right)+2$
$=3{x}^{2}+3x+3+2$
$=3{x}^{2}+3x+5$
Thus, $\left(h\circ f\right)\left(x\right)=3{x}^{2}+3x+5$