Question

Draw the graphs of f, g, and f + gon a common screen to illustrate graphical addition.

$f\left(x\right)=x^2$,$g\left(x\right)=\frac{1}{3}{x}^3$

Short answer

If f is a function with domain A, then the graph of f is the set of ordered pairs$\left\{\left(x,f\left(x\right)\right)|x\in A\right\}$plotted in a coordinate plane. In other words, the graph of f is the set of all points ( x, y ) such that $y=f\left(x\right)$; that is, the graph of f is the graph of the $y=f\left(x\right)$

Step-by-step solution

Step-1 – Definition of graph of a function

If f is a function with domain A, then the graph of f is the set of ordered pairs$\left\{\left(x,f\left(x\right)\right)|x\in A\right\}$plotted in a coordinate plane. In other words, the graph of f is the set of all points ( x, y ) such that $y=f\left(x\right)$; that is, the graph of f is the graph of the $y=f\left(x\right)$

Step-2 – Analyze the given data

Given $f\left(x\right)=x^2$,$g\left(x\right)=\frac{1}{3}{x}^3$

We first make a table of values.

x	$f\left(x\right)=x^2$	$g\left(x\right)=\frac{1}{3}{x}^3$	$f\left(x\right)+g\left(x\right)$
0	0	0	0
1	1	$\frac{1}{3}$	$\frac{4}{3}$
-1	1	$-\frac{1}{3}$	$\frac{2}{3}$

Step-3 – Graph

Plot the points given by the table and join them by a smooth curve to obtain the graph.

Since the domain of$f\left(x\right)=x^2$ is $D_f=\mathrm{ℝ}$, $g\left(x\right)=\frac{1}{3}{x}^3$is $D_g=\mathrm{ℝ}$, the domain of$f\left(x\right)+g\left(x\right)$ is given by$D_{f+g}=D_f\cap D_g=\mathrm{ℝ}$