4.1 Graphs of inverse functions

Intercepts

The general form of the inverse function of the form $y=ax+q$ is $y=\frac{1}{a}x-\frac{q}{a}$ .

By setting $x=0$ we have that the $y$ -intercept is $y_{int}=-\frac{q}{a}$ . Similarly, by setting $y=0$ we have that the $x$ -intercept is $x_{int}=q$ .

It is interesting to note that if $f\left(x\right)=ax+q$ , then $f^{-1}\left(x\right)=\frac{1}{a}x-\frac{q}{a}$ and the $y$ -intercept of $f\left(x\right)$ is the $x$ -intercept of $f^{-1}\left(x\right)$ and the $x$ -intercept of $f\left(x\right)$ is the $y$ -intercept of $f^{-1}\left(x\right)$ .

Exercises

1. Given $f\left(x\right)=2x-3$ , find $f^{-1}\left(x\right)$
2. Consider the function $f\left(x\right)=3x-7$ .
   1. Is the relation a function?
   2. If it is a function, identify the domain and range.
3. Sketch the graph of the function $f\left(x\right)=3x-1$ and its inverse on the same set of axes.
4. The inverse of a function is $f^{-1}\left(x\right)=2x-4$ , what is the function $f\left(x\right)$ ?

Inverse function of $y=a{x}^2$

The inverse relation, possibly a function, of $y=a{x}^2$ is determined by solving for $x$ as:

We see that the inverse ”function” of $y=a{x}^2$ is not a function because it fails the vertical line test. If we draw a vertical line through the graph of $f^{-1}\left(x\right)=\pm \sqrt{x}$ , the line intersects the graph more than once. There has to be a restriction on the domain of a parabola for the inverse to also be a function. Consider the function $f\left(x\right)=-x^2+9$ . The inverse of $f$ can be found by witing $f\left(y\right)=x$ . Then

If we restrict the domain of $f\left(x\right)$ to be $x\ge 0$ , then $\sqrt{9-x}$ is a function. If the restriction on the domain of $f$ is $x\le 0$ then $-\sqrt{9-x}$ would be a function, inverse to $f$ .

Exercises

1. The graph of $f^{-1}$ is shown. Find the equation of $f$ , given that the graph of $f$ is a parabola. (Do not simplify your answer)

   

2. $f\left(x\right)=2x^2$ .
   1. Draw the graph of $f$ and state its domain and range.
   2. Find $f^{-1}$ and, if it exists, state the domain and range.
   3. What must the domain of $f$ be, so that $f^{-1}$ is a function ?
3. Sketch the graph of $x=-\sqrt{10-y^2}$ . Label a point on the graph other than the intercepts with the axes.
4. 1. Sketch the graph of $y=x^2$ labelling a point other than the origin on your graph.
   2. Find the equation of the inverse of the above graph in the form $y=...$ .
   3. Now sketch the graph of $y=\sqrt{x}$ .
   4. The tangent to the graph of $y=\sqrt{x}$ at the point A(9;3) intersects the $x$ -axis at B. Find the equation of this tangent and hence or otherwise prove that the $y$ -axis bisects the straight line AB.
5. Given: $g\left(x\right)=-1+\sqrt{x}$ , find the inverse of $g\left(x\right)$ in the form $g^{-1}\left(x\right)=...$ .

Inverse function of $y=a^x$

The inverse function of $y=a{x}^2$ is determined by solving for $x$ as follows:

The inverse of $y={10}^x$ is $x={10}^y$ , which we write as $y=log\left(x\right)$ . Therefore, if $f\left(x\right)={10}^x$ , then $f^{-1}=log\left(x\right)$ .

The exponential function and the logarithmic function are inverses of each other; the graph of the one is the graph of the other, reflected in the line $y=x$ . The domain of the function is equal to the range of the inverse. The range of the function is equal to the domain of the inverse.

Exercises

1. Given that $f\left(x\right)={\left(\frac{1}{5}\right)}^x$ , sketch the graphs of $f$ and $f^{-1}$ on the same system of axes indicating a point on each graph (other than the intercepts) and showing clearly which is $f$ and which is $f^{-1}$ .
2. Given that $f\left(x\right)=4^{-x}$ ,
   1. Sketch the graphs of $f$ and $f^{-1}$ on the same system of axes indicating a point on each graph (other than the intercepts) and showing clearly which is $f$ and which is $f^{-1}$ .
   2. Write $f^{-1}$ in the form $y=...$ .
3. Given $g\left(x\right)=-1+\sqrt{x}$ , find the inverse of $g\left(x\right)$ in the form $g^{-1}\left(x\right)=...$
4. Answer the following questions:
   1. Sketch the graph of $y=x^2$ , labeling a point other than the origin on your graph.
   2. Find the equation of the inverse of the above graph in the form $y=...$
   3. Now, sketch $y=\sqrt{x}$ .
   4. The tangent to the graph of $y=\sqrt{x}$ at the point $A(9;3)$ intersects the $x$ -axis at $B$ . Find the equation of this tangent, and hence, or otherwise, prove that the $y$ -axis bisects the straight line $AB$ .

End of chapter exercises

1. Sketch the graph of $x=-\sqrt{10-y^2}$ . Is this graph a function ? Verify your answer.
2. $f\left(x\right)={\displaystyle \frac{1}{x-5}}$ ,
   1. determine the $y$ -intercept of $f\left(x\right)$
   2. determine $x$ if $f\left(x\right)=-1$ .
3. Below, you are given 3 graphs and 5 equations.

   

   1. $y={log}_{3}x$
   2. $y=-{log}_{3}x$
   3. $y={log}_3(-x)$
   4. $y=3^{-x}$
   5. $y=3^x$
   Write the equation that best describes each graph.
4. Given $g\left(x\right)=-1+\sqrt{x}$ , find the inverse of $g\left(x\right)$ in the form $g^{-1}\left(x\right)=...$
5. Consider the equation $h\left(x\right)=3^x$
   1. Write down the inverse in the form $h^{-1}\left(x\right)=...$
   2. Sketch the graphs of $h\left(x\right)$ and $h^{-1}\left(x\right)$ on the same set of axes, labelling the intercepts with the axes.
   3. For which values of $x$ is $h^{-1}\left(x\right)$ undefined ?
6. 1. Sketch the graph of $y=2x^2+1$ , labelling a point other than the origin on your graph.
   2. Find the equation of the inverse of the above graph in the form $y=...$
   3. Now, sketch $y=\sqrt{x}$ .
   4. The tangent to the graph of $y=\sqrt{x}$ at the point $A(9;3)$ intersects the $x$ -axis at $B$ . Find the equation of this tangent, and hence, or otherwise, prove that the $y$ -axis bisects the straight line $AB$ .