# Given h(x) = 2x + 4 and f(x) = 1/2x + 3, Evaluate the composite function f[h(x)]

Question
Composite functions
Given $$\displaystyle{h}{\left({x}\right)}={2}{x}+{4}$$ and $$\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{2}}{x}+{3}$$,
Evaluate the composite function f[h(x)]

2021-01-09
To evaluate the composite function f(h(x)), Substitute 2x+4 in the place of h(x)
$$\displaystyle{h}{\left({x}\right)}={2}{x}+{4}$$
$$\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{{2}{x}}}+{3}$$
$$\displaystyle{f}{\left[{h}{\left({x}\right)}\right]}={f{{\left({2}{x}+{4}\right)}}}$$
Now replace all x by 2x+4 in f(x)
$$\displaystyle{h}{\left({x}\right)}={2}{x}+{4}$$
$$\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{{2}{x}}}+{3}$$
$$\displaystyle{f}{\left[{h}{\left({x}\right)}\right]}={f{{\left({2}{x}+{4}\right)}}}$$
$$\displaystyle{F}{\left[{\left({2}{x}+{4}\right)}\right]}=\frac{{1}}{{{2}{\left({2}{x}+{4}\right)}}}+{3}=\frac{{1}}{{{4}{x}+{8}}}+{3}$$
Answer: $$\displaystyle{f}{\left[{h}{\left({x}\right)}\right]}=\frac{{1}}{{{4}{x}+{8}}}+{3}$$

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