Question Find the inverse function $f^{-1}(x)$ for the rational function $f(x) = \frac{7-5x}{5x+2}$ .

Studdy Solution

STEP 1

Assumptions
1. The function is given by $f(x)=\frac{7-5x}{5x+2}$ .
2. We want to find the inverse function, denoted as $f^{-1}(x)$ .
3. The function $f(x)$ is one-to-one, which means it has an inverse function.

STEP 2

To find the inverse function $f^{-1}(x)$ , we need to swap the roles of $x$ and $y$ in the original function and then solve for $y$ . Let's start by replacing $f(x)$ with $y$ .
$y = \frac{7-5x}{5x+2}$

STEP 3

Now, interchange $x$ and $y$ to get the equation in terms of $x$ and the new variable $y$ .
$x = \frac{7-5y}{5y+2}$

STEP 4

Next, we will solve for $y$ in terms of $x$ . Start by cross-multiplying to eliminate the fraction.
$x(5y+2) = 7-5y$

STEP 5

Distribute $x$ on the left side of the equation.
$5xy + 2x = 7 - 5y$

STEP 6

Move all terms involving $y$ to one side and the constant terms to the other side.
$5xy + 5y = 7 - 2x$

STEP 7

Factor out $y$ from the left side of the equation.
$y(5x + 5) = 7 - 2x$

STEP 8

Divide both sides by $(5x + 5)$ to solve for $y$ .
$y = \frac{7 - 2x}{5x + 5}$

STEP 9

Simplify the expression by factoring out a $-1$ from the numerator and a $5$ from the denominator.
$y = \frac{-(2x - 7)}{5(x + 1)}$

STEP 10

The expression for $y$ now represents the inverse function $f^{-1}(x)$ . Therefore, we can write the final answer as:
$f^{-1}(x) = \frac{-(2x - 7)}{5(x + 1)}$
This is the inverse function of $f(x)=\frac{7-5x}{5x+2}$ .

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