Chapter 1, Problem 4PS

To determine

To Graph: $f\left(x\right)=x$ & $g\left(x\right)=-x$ and find are they there own inverse also graph other linear functions that are their own inverse functions and find a formula for a family of linear functions that are their own inverse functions.

Answer to Problem 4PS

A line which is inverse of itself is $y=-x+c$ .

Explanation of Solution

Given information:

The functions $f\left(x\right)=x$ and $g\left(x\right)=-x$ are their own inverse functions.

Calculation:

Graph of $f\left(x\right)=x$

  $\begin{array}{cccccc}x& -4& -2& 0& 2& 4\\ f\left(x\right)& -4& -2& 0& 2& 4\end{array}$

Plot the points and join the dots

Graph of $f\left(x\right)=x$ is reflection of itself over the line $y=x$ .So it is inverse of own.

Now we plot graph of $g\left(x\right)=-x$

  $\begin{array}{cccccc}x& -4& -2& 0& 2& 4\\ g\left(x\right)& 4& 2& 0& -2& -4\end{array}$

  

Graph of $g\left(x\right)=-x$ is negative reflection of itself over the line $y=-x$ .So it is negative inverse of own.

Hence $f\left(x\right)=x$ is inverse of itself.

Now find other functions which are self inverse.

Let line be $y=mx+c\cdots \left(i\right)$ and let $x=m{y}^{\prime }+c\cdots \left(ii\right)$ be the inverse of that line.

From equation $\left(ii\right)y^{\prime }=\frac{x-c}{m}$

Now find the slope of the line by equating above two equations

Now if $y=y^{\prime }$ then

  $\begin{array}{l}mx+c=\frac{x-c}{m}\\ m^{2}x+cm=x-c\\ \left(m^2-1\right)x+c\left(m+1\right)=0\\ \left(m-1\right)\left(m+1\right)x+c\left(m+1\right)=0\\ \left(m+1\right)\left(xm-x+c\right)=0\\ \Rightarrow m=-1\end{array}$

Put $m=-1$ in equation $\left(i\right)$

  $y=-x+c$

Hence a line which is inverse of itself is $y=-x+c$