Question

given \\(f(x) = \frac{10x + 3}{-13x + 4}\\), find \\(f^{-1}(x)\\).

\\(f^{-1}(x) = \\)

Explanation:

Step1: Set function equal to y

$$y = \frac{10x + 3}{-13x + 4}$$

Step2: Swap variables x and y

$$x = \frac{10y + 3}{-13y + 4}$$

Step3: Multiply by denominator

$$x(-13y + 4) = 10y + 3$$

Step4: Expand the left side

$$-13xy + 4x = 10y + 3$$

Step5: Group terms with y

$$4x - 3 = 13xy + 10y$$

Step6: Factor out y

$$4x - 3 = y(13x + 10)$$

Step7: Solve for y

$$y = \frac{4x - 3}{13x + 10}$$

Answer:

\(f^{-1}(x) = \frac{4x - 3}{13x + 10}\)