Chapter 5.2, Problem 47AYU

In Problems 45-50, the graph of a one-to-one function $f$ is given. Draw the graph of the inverse function $f^{-1}$.

To determine

To find: The graph of the inverse function $f^{-1}$.

Answer to Problem 47AYU

Solution:

$f^{(-1)}=\left\{(0,-2),(1,0),(3,1)\right\}$

The graph of the inverse function is given as below:

Explanation of Solution

Given:

Calculation:

From the given graph of a one–to–one function $f$, we can write the points as in the following form.

i.e., $f=\left\{\left(-2,0\right),\left(0,1\right),\left(1,3\right)\right\}$

If the function $f$ is a set of ordered pairs $\left(x,y\right)$, then, the inverse function of $f$, denoted by $f^{-1}$, is the set of ordered pairs $\left(y,x\right)$.

The domain of the inverse of a relation is the same as the range of the original relation. In other words, the $y\text{-values}$ of the relation are the $x\text{-values}$ of the inverse.

Here, the given function $f$ is one–to–one function, so, $f^{-1}$ is also one–to–one function.

We know that $\left(x\right)=y\Rightarrow x=f^{-1}\left(y\right)$.

Therefore, the inverses of the above graph,

$f^{-1}\left(0\right)=-2$

$f^{-1}\left(1\right)=0$

$f^{-1}\left(3\right)=1$

$f^{(-1)}=\left\{(0,-2),(1,0),(3,1)\right\}$