Question

Question: Find $\mathit{f}\mathbf{}\mathbf{+}\mathbf{}\mathit{g}$and$\mathit{f}\mathit{g}$ for the functions f and g given in exercise 36.

Short answer

Answer:

The function$f+g$is$(f+g)\left(x\right)=x^2+x+3$

$fg$is$\left(fg\right)\left(x\right)=x^3+2x^2+x+2$

Step-by-step solution

Step 1:

Composition of f and g :$\mathbf{(}\mathit{f}\mathbf{\circ }\mathit{g}\mathbf{)}\mathbf{\left(}\mathit{a}\mathbf{\right)}\mathbf{=}\mathit{f}\mathbf{\left(}\mathit{g}\mathbf{\right(}\mathit{a}\mathbf{\left)}\mathbf{\right)}$

Step 2:

Given:

$$\begin{gathered} g:\mathrm{ℝ}\to \mathrm{ℝ}\text{and f}:\mathrm{ℝ}\to \mathrm{ℝ} \\ f\left(x\right)=x^2+1 \\ f\left(x\right)=x+2 \end{gathered}$$

Since f and g are both from$\mathrm{ℝ}\to \mathrm{ℝ}$, $f\circ g\text{and}g\circ f$are also functions from$\mathrm{ℝ}\to \mathrm{ℝ}$

Use the definition of composition:

$$\begin{gathered} (f+g)\left(x\right)=f\left(x\right)+g\left(x\right)=(x^2+1)+(x+2)=x^2+x+3 \\ \left(fg\right)\left(x\right)=f\left(x\right).g\left(x\right)=(x^2+1)(x+2)=x^3+2x^2+x+2 \end{gathered}$$

Hence, the solution is $f+g$is$(f+g)\left(x\right)=x^2+x+3$

$fg$is$\left(fg\right)\left(x\right)=x^3+2x^2+x+2$