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

Question
Composite functions
Given
$$\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}+{x}+{1}$$
h(x) = 3x + 2,
evaluate the composite function.

2020-11-02
By using the definition of a composite function,
$$\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}={f{{\left({h}{\left({x}\right)}\right)}}}$$
$$\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}={f{{\left({h}{\left({x}\right)}\right)}}}$$
=f(3x+2)
To find value of f(3x+2) replace x by 3x+2 in function f.
$$\displaystyle{f{{\left({3}{x}+{2}\right)}}}={\left({3}{x}+{2}\right)}^{{2}}+{\left({3}{x}+{2}\right)}+{1}$$
$$\displaystyle={9}{x}^{{2}}+{12}{x}+{4}+{3}{x}+{2}+{1}$$
$$\displaystyle={9}{x}^{{2}}+{15}{x}+{7}$$
Thus, $$\displaystyle{\left({f}\circ{h}\right)}{\left({x}\right)}={9}{x}^{{2}}+{15}{x}+{7}$$
Again by using the definition of a composite function,
$$\displaystyle{\left({h}\circ{f}\right)}{\left({x}\right)}={h}{\left({f{{\left({x}\right)}}}\right)}$$
$$\displaystyle{\left({h}\circ{f}\right)}{\left({x}\right)}={h}{\left({f{{\left({x}\right)}}}\right)}$$
$$\displaystyle={h}{\left({x}^{{2}}+{x}+{1}\right)}$$
To find $$\displaystyle{h}{\left({x}^{{2}}+{x}+{1}\right)}$$ replace $$\displaystyle{x}\in{h}{\left({x}\right)}{b}{y}{x}^{{2}}+{x}+{1}$$
$$\displaystyle{h}{\left({x}^{{2}}+{x}+{1}\right)}={3}{\left({x}^{{2}}+{x}+{1}\right)}+{2}$$
$$\displaystyle={3}{x}^{{2}}+{3}{x}+{3}+{2}$$
$$\displaystyle={3}{x}^{{2}}+{3}{x}+{5}$$
Thus, $$\displaystyle{\left({h}\circ{f}\right)}{\left({x}\right)}={3}{x}^{{2}}+{3}{x}+{5}$$

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