QuestionFind the function $g$ if $f(x)=\frac{1}{x}$ and $\left(\frac{f}{g}\right)(x)=\frac{x+1}{x^{2}-x}$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is given as $f(x)=\frac{1}{x}$ . The function $\left(\frac{f}{g}\right)(x)$ is given as $\left(\frac{f}{g}\right)(x)=\frac{x+1}{x^{}-x}$
3. We are asked to find the function $g(x)$

STEP 2

The function $\left(\frac{f}{g}\right)(x)$ is the division of function $f(x)$ by function $g(x)$ . So, we can write $\left(\frac{f}{g}\right)(x)$ as $\frac{f(x)}{g(x)}$ . $\frac{f(x)}{g(x)} = \frac{x+1}{x^{2}-x}$

STEP 3

We know that $f(x)=\frac{1}{x}$ . Substitute this into the equation.
$\frac{\frac{1}{x}}{g(x)} = \frac{x+1}{x^{2}-x}$

STEP 4

To find $g(x)$ , we need to isolate it on one side of the equation. We can do this by multiplying both sides of the equation by $g(x)$ and by $x$ .
$g(x) = \frac{x \cdot (x+1)}{x^{2}-x}$

STEP 5

implify the right side of the equation.
$g(x) = \frac{x^{2}+x}{x^{2}-x}$ So, the function $g(x)$ is $g(x) = \frac{x^{2}+x}{x^{2}-x}$ .

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