Question

In the following exercise, find

(a) $R\left(x\right)=f\left(x\right)+g\left(x\right)$

(b) $R\left(x\right)=f\left(x\right)-g\left(x\right)$:

$$\begin{gathered} f\left(x\right)=\frac{6x}{x^2-64} \\ g\left(x\right)=\frac{3}{x-8} \end{gathered}$$

Short answer

(a) $R\left(x\right)=\frac{3\left(3x+8\right)}{\left(x-8\right)\left(x+8\right)}$

(b)$R\left(x\right)=\frac{3}{x+8}$

Step-by-step solution

Step 1. Given information.

The following information is given:

$$\begin{gathered} f\left(x\right)=\frac{6x}{x^2-64} \\ g\left(x\right)=\frac{3}{x-8} \end{gathered}$$

Step 1. Finding (a) Rx=f(x)+g(x).

$R\left(x\right)=f\left(x\right)+g\left(x\right)$

Substituting the values

$R\left(x\right)=\frac{6x}{x^2-64}+\frac{3}{x-8}$

Factorise the denominators:

$R\left(x\right)=\frac{6x}{\left(x-8\right)\left(x+8\right)}+\frac{3}{\left(x-8\right)}$

Find out the LCD:

$\left(x-8\right)\left(x+8\right)$

Rewrite the expression with a denominator equal to the LCD:

$R\left(x\right)=\frac{6x}{\left(x-8\right)\left(x+8\right)}+\frac{3\left(x+8\right)}{\left(x-8\right)\left(x+8\right)}$

Simplify:

$$\begin{gathered} R\left(x\right)=\frac{6x+3x+24}{\left(x-8\right)\left(x+8\right)} \\ R\left(x\right)=\frac{9x+24}{\left(x-8\right)\left(x+8\right)} \\ R\left(x\right)=\frac{3\left(3x+8\right)}{\left(x-8\right)\left(x+8\right)} \end{gathered}$$

Step 3. Finding (b) Rx=fx-gx.

$R\left(x\right)=f\left(x\right)-g\left(x\right)$

By substituting the values and factorising:

$R\left(x\right)=\frac{6x}{\left(x-8\right)\left(x+8\right)}-\frac{3}{\left(x-8\right)}$

Finding the LCD:

$\left(x-8\right)\left(x+8\right)$

Rewriting the expression of $R\left(x\right)$:

$R\left(x\right)=\frac{6x}{\left(x-8\right)\left(x+8\right)}-\frac{3\left(x+8\right)}{\left(x-8\right)\left(x+8\right)}$

Simplifying:

$$\begin{gathered} R\left(x\right)=\frac{6x-3x-24}{\left(x-8\right)\left(x+8\right)} \\ R\left(x\right)=\frac{3x-24}{\left(x-8\right)\left(x+8\right)} \\ R\left(x\right)=\frac{3\left(x-8\right)}{\left(x-8\right)\left(x+8\right)} \end{gathered}$$

By cancelling the common factors from the numerator and the denominator:

$R\left(x\right)=\frac{3}{x+8}$