Chapter 10.3, Problem 24WE

To determine

Prove that squaring function $f(x)=x^2$ has no inverse function .Find the inverse of $g(x)=\sqrt{x}$ and explain why its inverse is not same as $f(x)=x^2$ .

Answer to Problem 24WE

  $g^{-1}(x)=x^2$

Domain = $\{x:x\ge 0\}$

Range = $\{y:y\ge 0\}$

Explanation of Solution

Calculation:

The function $f(x)=x^2$ have no inverse since it does not pass the horizontal line test .

A function has a inverse if and only if any horizontal line intersects the graph of the function atmost at one point.

Since $f(x)=x^2$ is a parabola which opens upward and is symmetric about x = 0 , any horizontal line y > 0 intersects it at two points.

Hence $f(x)=x^2$ has no inverse function.

Consider the function $g(x)=y=\sqrt{x}$ .

Domain = $\{x:x\ge 0\}$

Range = $\{y:y\ge 0\}$

Since the graph of the function is :

  

Clearly , it passes the horizontal line test . So, inverse of the function exists.

We find the inverse :

Interchange x and y values and solve for y :

  $\begin{array}{l}\Rightarrow x=\sqrt{y}\\ \Rightarrow y=x^2\\ \Rightarrow g^{-1}(x)=x^2\end{array}$

But the domain and range of the inverse are :

Domain = $\{x:x\ge 0\}$

Range = $\{y:y\ge 0\}$

Since the domain and range of the inverse functions are interchanged.

The inverse $g^{-1}(x)=x^2$ is different from $f(x)=x^2$ in the domain values.

Domain of $f(x)=x^2$ = {all real numbers}, whereas the domain of $g^{-1}(x)=x^2$ is $\{x:x\ge 0\}$ .