Chapter 3.4, Problem 44E

Solution

(a)

Whether the given statement is true or false.

False

Given information:

If $f\left(x\right)=x^n$ where n is a positive even integer, then the inverse of f is a function.

Concept used:

• The graph of the function $y=f\left(x\right)$ and its inverse $f^{-1}\left(x\right)$ are symmetric to the line $y=x$ .
• For any point $\left(a,b\right)$ on the graph of $f\left(x\right)$ , the corresponding point on the graph of inverse function $f^{-1}\left(x\right)$ is $\left(b,a\right)$ .
• If a function is one-to-one, then it has an inverse.

Calculation:

Note that the graph of the function of the form $f\left(x\right)=x^n$ , where n is an even integer, is not one-to-one and hence does not have an inverse.

The graph of an even degree polynomial is either both ends upward or both ends open downwards and it does not satisfy the horizontal line test. While in a restricted domain the even degree polynomials do have inverse.

Thus, the given statement is false.

(b)

Whether the given statement is true or false.

True

Given information:

If $f\left(x\right)=x^n$ where n is a positive odd integer, then the inverse of f is a function.

Concept used:

• The graph of the function $y=f\left(x\right)$ and its inverse $f^{-1}\left(x\right)$ are symmetric to the line $y=x$ .
• For any point $\left(a,b\right)$ on the graph of $f\left(x\right)$ , the corresponding point on the graph of inverse function $f^{-1}\left(x\right)$ is $\left(b,a\right)$ .
• If a function is one-to-one, then it has an inverse.

Calculation:

Note that the graph of the function of the form $f\left(x\right)=x^n$ , where n is an odd integer, is a one-to-one and hence does have an inverse.

The ends of the graph of an odd degree polynomial opens in opposite directions and it does satisfy the horizontal line test.

Thus, the given statement is true.