If f ( x ) = \frac { x - 3} { x } and g(x) = 5x-4, what is the domain of (f*g)(x)?

Mar 29, 2018

$\left\{x \in R | x \ne \frac{4}{5}\right\}$

Explanation:

First figure out what $\left(f \cdot g\right) \left(x\right)$ is to do this just put the $g \left(x\right)$ function into both of the x spots in $f \left(x\right)$

$\left(f \cdot g\right) \left(x\right) = \frac{5 x - 4 - 3}{5 x - 4}$ so $\left(f \cdot g\right) \left(x\right) = \frac{5 x - 7}{5 x - 4}$

We note that for a rational function basically $\frac{1}{x}$ when the denominator is equal to 0 there is no output

So we have to figure out when $5 x - 4 = 0$

$5 x = 4$ so $x = \frac{4}{5}$

So the domain is all the reals apart from $x = \frac{4}{5}$
$\left\{x \in R | x \ne \frac{4}{5}\right\}$