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The graph of $$f$$ is given. Draw the graphs of the following functions:

$$y=f{(x)-3} \quad y=f(x+1)$$

$$y=\frac{1}{2} f(x) \quad y=-f(x)$$

Find the Functions: $$F(x)=3 x+5 g(x)=x^{2}+x$$

(a) $$f^{0} g \quad(c) g_{0} f$$

(d) $$f_{0} f \quad(\mathrm{b}) g_{0} g$$

(a)fog $$=f[g(x)]=F\left[x^{2}+x\right]=3\left[x^{2}+x\right]+5$$

$$=3 x^{2}+3 x+5$$ Domain R

(b) $$g_{0} g=g[g(x)]=\left(x^{2}+x\right)^{2}+\left(x^{2}+x\right)$$

$$=x^{4}+2 x^{3}+x^{2}+x^{2}+x=x^{4}+2 x^{3}+2 x^{2}+x$$ Domain R

$$(c) g{o} f=g[f(x)]=(3 x+5)^{2}+(3 x+5)$$

$$=9 x^{2}+30 x+25+3 x+5=9 x^{2}+33 x+30$$ Domain R

(d) $$f o f=f[f(x)]=3(3 x+5)+5=9 x+15+5=9 x+20$$ Domain R

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