Chapter 9.4, Problem 45HP

To determine

To prove:

The function $f\left(x\right)=\frac{x}{a-b}+c$ is a rational function

Explanation of Solution

Given information:

A rational function has an equation of the form $f\left(x\right)=\frac{a\left(x\right)}{b\left(x\right)}$ , where function $a\left(x\right)\text{ and }b\left(x\right)$ are polynomial functions and $b\left(x\right)\ne 0$

Proof:

Find the common denominator and add the two expressions

  $\begin{array}{l}f\left(x\right)=\frac{x}{a-b}+c\\ f\left(x\right)=\frac{x}{a-b}+\frac{c\left(a-b\right)}{\left(a-b\right)}\\ f\left(x\right)=\frac{x+c\left(a-b\right)}{a-b}\end{array}$

A rational function has an equation of the form $f\left(x\right)=\frac{a\left(x\right)}{b\left(x\right)}$ and the value of $a\left(x\right)\text{ and }b\left(x\right)$ are found. Thus,function $a\left(x\right)\text{ and }b\left(x\right)$ are polynomials and $b\left(x\right)\ne 0$

Hence it is proved that the function $f\left(x\right)$ is a rational function.