Chapter 2.2, Problem 43E

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.

  $k=3,a=\frac{1}{4}$ .

The function is increasing in 1^st quadrant.

For $x<0$ , the function is undefined.

Given:

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

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

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

Hence, $k=3,a=\frac{1}{4}$ .

Since, the power of this function is $\frac{1}{4}$ , which is an even radical. Hence, it lies only in the 1^st quadrant and it is increasing.

Also. Since, the order of the radical is even, hence for $x<0$ , the function is undefined.

Below, is the graph for the function.