Inverse Functions

Prove, using the composition of functions, 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 analytical expression of the inverse function of the linear function $f(x) = 3x - 5$.

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

Calculate the inverse function of $h(x) = \sqrt{2x - 1}$, indicating the domain of the new function obtained.

Given the graph of the exponential function $f(x) = 2^x$, draw the coordinate axes and the line $y = x$ in your notebook, 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.

Obtain the inverse function of the exponential function $f(x) = 2^{x-3}$.

Find the inverse function of the logarithmic function $g(x) = \log_3(x) + 1$.

The sound intensity level in decibels (dB) perceived by the human ear is calculated using the function $\beta(I) = 10 \log_{10} \left( \frac{I}{I_0} \right)$, where $I$ is the acoustic intensity in $\text{W/m}^2$ and $I_0 = 10^{-12} \text{ W/m}^2$ is the threshold of hearing intensity. Find the inverse function $I(\beta)$ that determines the intensity as a function of the decibels. If the human threshold of pain is situated at $120 \text{ dB}$, what is the corresponding acoustic intensity?

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

The monthly production cost in dollars of $x$ units of an electronic component is given by the function $C(x) = \frac{120x + 500}{x + 2}$. Find the inverse function $C^{-1}(x)$ and interpret what this new function represents in the context of the problem.

Prove algebraically that if $f(x) = \frac{x^3 + 1}{x^3 - 1}$, then its inverse function is given by the expression $f^{-1}(x) = \sqrt[3]{\frac{x+1}{x-1}}$. Next, justify by solving an equation why the graph of $f(x)$ never intersects the graph of its inverse function.

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

Calculate the inverse function of the following piecewise function:

$f(x) = \begin{cases} 2x + 1 & \text{if } x < 0 \\ x^2 + 1 & \text{if } x \ge 0 \end{cases}$

Prove analytically that the domain of $f^{-1}$ coincides exactly with the range of $f$.

The relationship that allows converting a given temperature in degrees Celsius ($C$) to degrees Fahrenheit ($F$) is $F(C) = \frac{9}{5}C + 32$. Find the inverse function $C(F)$ and use it to calculate how many degrees Celsius correspond to $68$ degrees Fahrenheit.

In a laboratory culture, the growth of a population of bacteria is modeled by the function $P(t) = 150 \cdot e^{0.2t}$, where $t$ is measured in hours. Determine the inverse function $t(P)$ and use it to calculate exactly how much time must elapse for the population to reach $3000$ bacteria.

The spread of a computer virus in a closed network is modeled by the logistic function $N(t) = \frac{5000}{1 + 49e^{-0.5t}}$, where $N(t)$ is the number of infected computers and $t$ represents the days elapsed since the initial detection. Determine the inverse function $t(N)$. What is the meaning of the domain of this inverse function in the context of the virus's propagation limit?

The annual depreciation of industrial equipment is modeled by the function $V(t) = 12000 \cdot (0.8)^t$, where $V$ is the current value in dollars and $t$ is the time elapsed in years since its purchase. Find the inverse function $t(V)$ and indicate exactly how many years have passed if the equipment is valued today at $\$6144$.

The concentration $C(t)$ of a reactant in a second-order chemical reaction evolves according to the law $C(t) = \frac{C_0}{1 + k \cdot C_0 \cdot t}$, where $C_0$ is the initial concentration and $k$ is the rate constant. Make time $t$ the subject to find the inverse function $t(C)$. Use it to prove that the half-life (the time necessary for the concentration to halve, $C = \frac{C_0}{2}$) is inversely proportional to the initial concentration $C_0$.

Knowing that $f(x) = x^3 + 2x + 1$, calculate the numerical value of $f^{-1}(4)$ without needing to find the complete algebraic expression of the inverse function.

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

Knowing that the rational function $f(x) = \frac{ax + b}{cx - 1}$ is its own inverse function (that is, $f^{-1}(x) = f(x)$) and that its graph passes through the point $(2, 5)$, set up a system of equations and determine the numerical values of the parameters $a$, $b$ and $c$.

The graph of the function $f(x) = x^2 - 2$ restricted to the domain $x \ge 0$ is shown. Draw the line $y = x$ and the graph of the inverse function $f^{-1}(x)$ in your notebook. Analytically determine the exact point of intersection between both curves.

Let the radical function be $f(x) = \sqrt{x+6}$. Calculate its inverse function $f^{-1}(x)$, clearly indicating its domain. Analytically determine the points of intersection between the graphs of $f(x)$ and $f^{-1}(x)$, knowing that by symmetry these points must lie on the line $y = x$.

Given the functions $f(x) = \frac{x}{x-2}$ and $g(x) = \sqrt{x+1}$, obtain step by step the analytical expression of the inverse function of the composition, that is, $(f \circ g)^{-1}(x)$, and determine the domain of validity of this new function.