# For the functions f(x)=10x-8 \text{and} g(x)=5x+7, find the following. (a)(f+g)(x) (b)(f-g)(x)

For the functions $$\displaystyle{f{{\left({x}\right)}}}={10}{x}-{8}\text{and}{g{{\left({x}\right)}}}={5}{x}+{7}$$, find the following.
$$\displaystyle{\left({a}\right)}{\left({f}+{g}\right)}{\left({x}\right)}$$
$$\displaystyle{\left({b}\right)}{\left({f}-{g}\right)}{\left({x}\right)}$$

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Step 1
Let Given function $$\displaystyle{f{{\left({x}\right)}}}={10}{x}-{8}{\quad\text{and}\quad}{g{{\left({x}\right)}}}={5}{x}+{7}$$
$$\displaystyle{\left({a}\right)}{\left({f}+{9}\right)}{\left({x}\right)}$$
we know that, $$\displaystyle{\left({f}+{9}\right)}{\left({x}\right)}={f{{\left({x}\right)}}}+{9}{\left({x}\right)}$$
$$\displaystyle\Rightarrow{\left({f}+{9}\right)}{\left({x}\right)}={f{{\left({x}\right)}}}+{9}{\left({x}\right)}$$
$$\displaystyle={\left({10}{x}-{8}\right)}+{\left({5}{x}+{7}\right)}$$
$$\displaystyle{\left({f}+{9}\right)}{\left({x}\right)}={15}{x}-{1}$$
Step 2
$$\displaystyle{\left({b}\right)}{\left({f}-{9}\right)}{\left({x}\right)}$$
we know that, $$\displaystyle{\left({f}-{9}\right)}{\left({x}\right)}={f{{\left({x}\right)}}}-{9}{\left({x}\right)}$$
$$\displaystyle{\left({f}-{9}\right)}{\left({x}\right)}={f{{\left({x}\right)}}}-{9}{\left({x}\right)}$$
$$\displaystyle={\left({10}{x}-{8}\right)}-{\left({5}{x}+{7}\right)}$$
$$\displaystyle={10}{x}-{8}-{5}{x}-{7}$$
$$\displaystyle{\left({f}-{9}\right)}{\left({x}\right)}={5}{x}-{15}$$