Chapter 2.2, Problem 84E

To determine

To describe whether the graph represent a function of $x$ .

Answer to Problem 84E

The equation, $x=y^2$ does not define $y$ as a function $x$ because the equation gives two value of $y$ for each value of $x$ .

The equation, $x=y^3$ does not define $y$ as a function $x$ because the equation gives two real value of $y$ for each value of $x$ .

For all rational value of $n$ , the graph $x=y^n$ is the graph of a function of $x$ .

Explanation of Solution

Given information :

The function is $f\left(x\right)=x^n$ for every integer $n$ .

Concept used:

The graph of the function $y=f\left(x\right)$ represent a function of $x$ if and only if there is only one value of $y$ for each value of $x$ .

For the function, $x=y^2$ ,

Solve for $y$ in terms of $x$ ,

  $\begin{array}{l}x=y^2\\ y=\pm \sqrt{x}\end{array}$

The last equation gives two value of $y$ that is $+\sqrt{x}\text{ or }-\sqrt{x}$ for a given value of $x$ .

Thus, the equation does not define $y$ as a function $x$ .

Similarly,

For the function, $x=y^3$ ,

Solve for $y$ in terms of $x$ ,

  $\begin{array}{l}x=y^3\\ y=\sqrt[3]{x}\end{array}$

The last equation gives two real valueand one complex value of $y$ for a given value of $x$ .

Thus, the equation does not define $y$ as a function $x$ .

If the function is in the form of $y=x^n$ , then only the equation define $y$ as a function $x$ .

Now,

The function in the form of $x=y^n$ will be the equation which define $y$ as a function $x$ if and only if $n$ is a rational number.

Assume, $n=\frac{1}{2}$ , then the function is thus obtained is $x=y^{\frac{1}{2}}$ ,

Solve for $y$ in terms of $x$ ,

  $\begin{array}{l}x=y^{\frac{1}{2}}\\ y=x^2\end{array}$ [Square both sides]

The last equation gives only one real value of $y$ for a given value of $x$ .

Thus, the equation defines $y$ as a function $x$ .

Hence,

For all rational value of $n$ , the graph $x=y^n$ is the graph of a function of $x$ .