Functions Study Guide Question

Question

I just have a quick question because I couldn't find the answer to this question anywhere in my textbook or any formula, and I finished everything else I was supposed to.

If $$f(x) = \frac{x}{3x-1},$$ then $$\frac{f(x)-f(a)}{x-a} = ?$$

Answer 1

$\frac{(\color{blue}{f(x)})-(\color{green}{f(a)})}{x-a}$

$\frac{(\color{blue}{\frac{x}{3x-1}})-(\color{green}{\frac{a}{3a-1}})}{x-a}$

$\frac{\frac{\color{blue}{x}(\color{green}{3a-1})-\color{green}{a}(\color{blue}{3x-1})}{(\color{blue}{3x-1})(\color{green}{3a-1})}}{x-a}$

This can also be written as $\frac{\color{blue}{x}(\color{green}{3a-1})-\color{green}{a}(\color{blue}{3x-1})}{(x-a)(\color{blue}{3x-1})(\color{green}{3a-1})}$

Distribute a bit on top and somethings will zero out on top.

$\frac{({\color{blue}{x}\color{green}{3a}}-\color{blue}{x}\color{green}{1})-(\color{green}{a}\color{blue}{3x}-\color{green}{a}\color{blue}{1})}{(x-a)(\color{blue}{3x-1})(\color{green}{3a-1})}$

$\frac{(-\color{blue}{x}\color{green}{1})-(-\color{green}{a}\color{blue}{1})}{(x-a)(\color{blue}{3x-1})(\color{green}{3a-1})}$

$\frac{-x+a}{(x-a)(3x-1)(3a-1)}$

Now notice $-x+a=-(x-a)$ And see if you can continue from there.