Chapter 1.5, Problem 51E

(a)

To determine

To explain:The convincing argument that $f$ is a one-to-one.

Answer to Problem 51E

By the definition of one-to-one function, the function $f\left(x\right)=mx+b$ is one-to-one.

Explanation of Solution

Given information:The function is $y=f\left(x\right)=mx+b$ , and $m\ne 0$ .

It is known that a function is one-to-one on its domain if $f\left(x\right)=f\left(y\right)$ implies $x=y$ .

For any $x$ , and $y$ in the domain of the function suppose that $f\left(x\right)=f\left(y\right)$ . Then

  $\begin{array}{c}f\left(x\right)=mx+b\\ f\left(y\right)=my+b\end{array}$

As it is known that $f\left(x\right)=f\left(y\right)$ . So,

  $\begin{array}{c}mx+b=my+b\\ mx=my\\ x=y\end{array}$

Therefore, by the definition of one-to-one function, the function $f\left(x\right)=mx+b$ is one-to-one.

(b)

To determine

To find:The formula for the inverse of $f$ , and the relation between the slope of $f$ and $f^{-1}$ .

Answer to Problem 51E

The inverse of $f$ is $f^{-1}\left(x\right)=\frac{x-b}{m}$ , and slopes are reciprocal to each other.

Explanation of Solution

Given information:The function is $y=f\left(x\right)=mx+b$ , and $m\ne 0$ .

Calculation:

Interchange the $x$ and $y$ in the equation $y=mx+b$ , and solve for $y$ .

  $\begin{array}{c}x=my+b\\ my=x-b\\ y=\frac{x-b}{m}\end{array}$

Substitute $f^{-1}\left(x\right)$ for $y$ in the above equation.

  $f^{-1}\left(x\right)=\frac{x-b}{m}$

As the slope of $f$ is $m$ , and the slope of $f^{-1}$ is $\frac{1}{m}$ . It can be concluded that the slopes are multiplicative inverse of each other.

Therefore, the inverse of $f$ is $f^{-1}\left(x\right)=\frac{x-b}{m}$ , and slopes are reciprocal to each other.

(c)

To determine

To write:The conclusion about the graphs of the inverse of the functions if the graph of two functions are parallel line with non-zero slope.

Answer to Problem 51E

The graphs of the inverse will be parallel lines with non-zero slope.

Explanation of Solution

Given information:It is given that the graphs of two functions are parallel lines with non-zero slope.

If the original functions hasslope $m$ , then the graph of inverse of these functions will have the slope $\frac{1}{m}$ .

Therefore, the graphs of the inverse will be parallel lines with non-zero slope.

(d)

To determine

To write:The conclusion about the graphs of the inverse of the functions.

Answer to Problem 51E

The graphs of the inverse will be perpendicular lines with non-zero slope.

Explanation of Solution

Given information:It is given that the graphs of two functions are perpendicular lines with non-zero slope.

The lines are perpendicular so the original function will have the slopes $m$ and $-\frac{1}{m}$ respectively. So, the slopes of the inverse will be $\frac{1}{m}$ , and $-m$ respectively.

Since each of $\frac{1}{m}$ and $m$ is the negative reciprocal of the each other so the graph of the inverse will be perpendicular lines with non-zero slopes.

Therefore, the graphs of the inverse will be perpendicular lines with non-zero slope.