Inverse function

The quadratic function $f(x) = x^2 - 6x + 8$ does not have an inverse function on all of $\mathbb{R}$ because it is not injective. Restrict its domain to the interval $[3, +\infty)$ and calculate the analytical expression for its inverse function on that interval.

Find the analytical expression for the inverse function of the affine function $f(x) = 3x - 5$.

Determine the inverse function of the rational function $g(x) = \frac{4}{x+2}$.

Using function composition, check whether $f(x) = \frac{x+1}{2}$ and $g(x) = 2x - 1$ are inverse functions. You must verify that $(f \circ g)(x) = x$ and that $(g \circ f)(x) = x$.

Find the value of the parameter $k$ such that the function $f(x) = \frac{3x + 2}{x + k}$ is its own inverse, that is, satisfying the condition $f(f(x)) = x$ for any value in its domain.

Given the graph of the exponential function $f(x) = 2^x$, draw the coordinate axes and the line $y = x$ in your exercise book, and sketch the graph of its inverse function $f^{-1}(x)$. Identify the coordinates of at least three points through which $f^{-1}(x)$ passes.

You can do it on paper and send a photo with your answer.

Given the rational function $f(x) = \frac{2x - 1}{x + 3}$, find its inverse function $f^{-1}(x)$. Then, calculate the domain of $f(x)$ and the domain of $f^{-1}(x)$, and explain the relationship between these domains and the ranges of both functions.