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.

The fuel consumption of a car driving along an Irish dual carriageway at a constant speed is modeled by the function $C(v) = 4.2 + 0.0003 \times (v - 80)^2$, where $v$ is the vehicle speed in $\text{km/h}$, restricted to the regulatory interval $[80, 120]$, and $C(v)$ is the consumption in liters per $100\text{ km}$ ($\text{L}/100\text{ km}$).

1. a) Mathematically justify, by analyzing the monotonicity of $C(v)$ on the interval $[80, 120]$, why the function is injective on this domain and why this guarantees the existence of its inverse function.
2. b) Obtain the analytical expression of the inverse function $v(C)$ and state with justification its domain of definition and interpretation in the context of driving.
3. c) If the trip computer indicates that the average consumption on a flat section was $4.68\text{ L}/100\text{ km}$, use the function $v(C)$ to determine the exact speed at which the vehicle was traveling.

Consider the real piecewise-defined function:

where $b$ is a real parameter.

1. a) Find the vertex of the parabola corresponding to the second branch and determine its range for $x > 2$. Relying on the horizontal line test, deduce with justification what condition the maximum value of the first branch must satisfy for $f$ to be injective on all of $\mathbb{R}$, and write the inequality that the parameter $b$ must satisfy.
2. b) In the limiting case where the function turns out to be continuous ($b = -1$), justify why $f$ is bijective from $\mathbb{R}$ to $\mathbb{R}$ and deduce the analytical expression of its inverse function $f^{-1}(x)$, stating precisely the domain of each of its branches.
3. c) Without needing to calculate $f(x)$ beforehand, use the properties of composition and the function obtained in the previous part to solve the equation $(f^{-1} \circ f)(x) = f^{-1}(7)$.

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

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

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

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 number of users of a new messaging app grows according to the model $U(t) = 150 \times e^{0.2t}$, where $t$ is measured in weeks and $U(t)$ in thousands of users. Determine the inverse function $t(U)$ and use it to calculate the exact time that must elapse for the app to reach $3000$ thousand users (three million).

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 sound 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 decibels. If the human pain threshold is at $120 \text{ dB}$, what is the corresponding sound intensity?

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.

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 points $(2, 5)$ and $(0, -3)$, set up a system of equations and determine the numerical values of the parameters $a$, $b$, and $c$.

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.

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

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.