Chapter 2.2, Problem 31E

Solution

Describe how to obtain the graph for the monomial function $f\left(x\right)=\frac{2}{3}{x}^4$ from the graph of $g\left(x\right)=x^n$ with the same power n . State whether f is even or odd. Graph the function.

The power function $g\left(x\right)=x^4$ is shrunk vertically by $\frac{2}{3}$ units to obtain the graph of the given monomial function.

The given function is an even function.

Given:

  $f\left(x\right)=\frac{2}{3}{x}^4$ .

Concept Used:

• If any constant suppose c is multiplied with the function f , then the transformation rules for $cf\left(x\right)$ are
• If $c>1$, then the function will be stretched vertically by ‘c’ units.
• If $0<c<1$, then the function will shrink vertically by ‘c’ units.
• A function $f(x)$ is an even function if $f\left(-x\right)=f\left(x\right)$ .
• A function $f(x)$ is an odd function if $f\left(-x\right)=-f\left(x\right)$ .

Calculation:

The given function is $f\left(x\right)=\frac{2}{3}{x}^4$ .

Here, $\frac{2}{3}$ is multiplied with the power function. Since, it is in between $0$ and $1$ .

Hence, from the transformation rule, this function is shrink vertically by $\frac{2}{3}$ units.

Now,

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

  $f\left(-x\right)=\frac{2}{3}{x}^4$

  $f\left(-x\right)=f\left(x\right)$ .

Hence, it is an even function.

The graph of the function is shown below.