Question

Find f + g and fg for the functions f and g given in exercise 36.

Short answer

The function $f+g$is data-custom-editor="chemistry" $(f+g)\left(x\right)=x^2+x\mathit{+}\mathit{3}$

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

Step-by-step solution

Step 1:

Composition of f and g :$(f\circ g)\left(a\right)=f\left(g\right(a\left)\right)$

Step 2:

Given:

$g:\mathrm{ℝ}\to \mathrm{ℝ}andf:\mathrm{ℝ}\to \mathrm{ℝ}$

$\begin{array}{r}f\left(x\right)=x^2+1\\ f\left(x\right)=x+2\end{array}$

Since f and g are both from role="math" localid="1668671685751" $\mathrm{ℝ}\to \mathrm{ℝ},f\circ gandg\circ f$, are also functions from$\mathrm{ℝ}\to \mathrm{ℝ}$

Use the definition of composition:

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

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

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