Chapter 2.2, Problem 44E

Solution

State the values of the constants k and a for the function $f\left(x\right)=k\cdot x^a$ .Before using the grapher, describe the portion of the curve that lies in Quadrant I or IV. Determine whether f is even, off, or undefined for $x<0$ . Graph the function to see whether it matches the description.

  $k=3,a=\frac{1}{4}$ .

In 4^th quadrant the graph will be decreasing.

It is an even function.

Given:

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

Concept Used:

• A power function can be represented in the form $P\left(x\right)=k\cdot x^a$, here k is the constant of variation and a is the power.
• 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 $f\left(-x\right)=-f\left(x\right)$ .
• If the order of radical is even, then the function is undefined for $x<0$ .

Calculation:

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

It is in the form of $P\left(x\right)=k\cdot x^a$ .

Hence, $k=-4,a=\frac{2}{3}$ .

Since, there is a negative sign before the function, hence the parent function is reflected about the x-axis. Therefore, in 4^th quadrant the graph will be decreasing.

Now,

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

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

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

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

Hence, it is an even function.

Below, is the graph for the function.