Chapter 1.5, Problem 89E

To determine

To check: Whether a function that represents the graph of $f\left(x\right)=x^2$ shifted three units to the right is $f\left(g\left(x\right)\right)$ or not, where $g\left(x\right)=x+3$ .

Answer to Problem 89E

No, a function that represents the graph of $f\left(x\right)=x^2$ shifted three units to the right is not $f\left(g\left(x\right)\right)$ .

Explanation of Solution

Given information:

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

Replace the variable $x$ with $x-3$ to shift the graph of function by three units to the right.

  $\begin{array}{c}f\left(x\right)=x^2\\ f\left(x-3\right)={\left(x-3\right)}^2\end{array}$

Now simplify the composition.

  $\begin{array}{c}f\left(g\left(x\right)\right)=f\left(x+3\right)\\ ={\left(x+3\right)}^2\end{array}$

The function ${\left(x-3\right)}^2$ is not equal to the function ${\left(x+3\right)}^2$ .

Therefore, a function that represents the graph of $f\left(x\right)=x^2$ shifted three units to the right is not $f\left(g\left(x\right)\right)$ .