Chapter 6.2, Problem 50PPS

(a)

To determine

To Write:

A function that converts nautical miles to kilometers.

Answer to Problem 50PPS

  $K(m)=\text{1}\text{.852}m$

Explanation of Solution

Given Information:

There are approximately 1.852 kilometers in a nautical mile.

Calculation:

  $\begin{array}{l}Given\\ \text{1 nautical mile = 1}\text{.852 kilometeres}\\ \text{Let there are 'm' nautical miles }\text{.}\\ \text{So, there will be :}\\ \text{m nautical miles = 1}\text{.852m kilometeres }\\ \text{Hence , the function that converts nautical miles }\\ \text{to kilometeres is :}\\ K(m)=\text{1}\text{.852}m\end{array}$

(b)

To determine

To Find:

The inverse of the function that converts kilometers back to nautical miles

Answer to Problem 50PPS

  ${\text{K}}^{-1}(m)=\frac{m}{1.852}$

Explanation of Solution

Given Information:

From part (a) , we have $K(m)=\text{1}\text{.852}m$

Calculation:

  $\begin{array}{l}Given\\ K(m)=\text{1}\text{.852}m\\ \text{Replace K(m) by y and m by x:}\\ y=\text{1}\text{.852x}\\ \text{Replace y by x and x by y :}\\ x=\text{1}\text{.852y}\\ Solve\text{ for y:}\\ y=\frac{x}{1.852}\mathrm{............}[\text{divide each side by 1}\text{.852]}\\ \text{⇒}{\text{K}}^{-1}(m)=\frac{m}{1.852}\end{array}$

(c)

To determine

To Verify:

Using composition , verify that the functions are inverses.

Explanation of Solution

Given Information:

From part (a) , we have

  $K(m)=\text{1}\text{.852}m$

From part (b) , we have

  ${\text{K}}^{-1}(m)=\frac{m}{1.852}$

Concept Used :

 A function  g is the inverse of a function  f if whenever

  $\begin{array}{l}y=f(x)\text{ then }x=g(y)\\ \text{That is ,}\\ \text{f and g are inverse functions if :}\\ \left[f\circ g\right](x)=x\\ and\\ \left[g\circ f\right](x)=x\end{array}$

Calculation:

  $\begin{array}{l}Given\\ K(m)=\text{1}\text{.852}m\\ {\text{K}}^{-1}(m)=\frac{m}{1.852}\\ \left[K\circ K^{-1}\right](m)=K\left[K^{-1}(m)\right]\\ =K\left(\frac{m}{1.852}\right)\\ =\text{1}\text{.852}\left(\frac{m}{1.852}\right)=m\\ \left[K^{-1}\circ K\right](m)=K^{-1}\left[K(m)\right]\\ =K^{-1}\left(\text{1}\text{.852}m\right)\\ =\frac{\text{1}\text{.852}m}{1.852}=m\\ \text{So, }\\ \left[K\circ K^{-1}\right](m)=\left[K^{-1}\circ K\right](m)=m\\ \text{Hence , the functions are inverses}\text{.}\end{array}$