Exercises: Inverse function

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.

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.

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 $(g \circ f)(x) = x$.

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

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

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.

Seán has to find the inverse function of $f(x) = \frac{3x - 5}{2}$. On his exam, he writes the following solution:

"Since the inverse is denoted by $f^{-1}(x)$, I apply the property of negative exponents by inverting the fraction:

1. a) Identify and explain in detail the conceptual error made by Seán, clearly distinguishing between the reciprocal or multiplicative inverse of a number and the inverse function with respect to composition.
2. b) Correct the exercise by carrying out the appropriate procedure step by step to find the true expression for $f^{-1}(x)$.
3. c) Check that your result is correct by verifying that $(f \circ f^{-1})(x) = x$.

At a Met Éireann weather station located at sea level, a temperature of $24\text{ }^\circ\text{C}$ is recorded. In the adjacent mountainous region, the temperature decreases linearly at a rate of $0.65\text{ }^\circ\text{C}$ for every $100\text{ m}$ of altitude ($h$, expressed in hundreds of meters), according to the model $T(h) = 24 - 0.65h$.

1. a) Before carrying out any analytical calculation, estimate whether finding a temperature of $11\text{ }^\circ\text{C}$ will require climbing higher or lower than $1500\text{ m}$ of altitude ($h = 15$) and briefly justify your qualitative reasoning.
2. b) Solve for the variable $h$ in the equation to obtain the inverse function $h(T)$, interpret what quantity this function calculates, and use it to find the exact altitude (in meters) at which the temperature is $11\text{ }^\circ\text{C}$.
3. c) Compare the obtained value with your previous estimate and assess whether the result is consistent with the physical behavior of the model.

The monthly production cost in euros 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.

The relationship used to convert a temperature given 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.

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.

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 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 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.

The annual depreciation of industrial equipment is modelled by the function $V(t) = 12000 \times (0,8)^t$, where $V$ is the current value in euros and $t$ is the time elapsed in years since its purchase. Find the inverse function $t(V)$ and state how many years have passed exactly if the equipment is valued today at $6144$ euros.

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 exercise book. Determine analytically the exact point of intersection between both curves.

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

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.

The aim is to determine the value of the coefficients of a linear fractional (homographic) function of the form $f(x) = \frac{ax + 4}{2x - 3}$ (with $x \neq 1.5$) so that it simultaneously satisfies two conditions: its graph must pass through the point $(1, -7)$ and it must be identical to its own inverse function, that is, $f(x) = f^{-1}(x)$ on its entire domain.

1. a) Set up and solve the equation that allows you to calculate the real parameter $a$ from the point through which the graph passes.
2. b) With the found value of $a$, calculate the algebraic expression of $f^{-1}(x)$ by solving for the independent variable and verify whether the equality $f(x) = f^{-1}(x)$ indeed holds.
3. c) Reason geometrically what symmetry with respect to the line $y = x$ the graph of any function that coincides with its own inverse must possess.

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?

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 composite function, that is, $(f \circ g)^{-1}(x)$, and determine the domain of validity of this new function.

Calculate the inverse function of the following piecewise function:

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

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

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 $c$.

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

Let $f(x) = \sqrt{x+6}$ be the radical function. Calculate its inverse function $f^{-1}(x)$, clearly stating 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 $y = x$.

The spread of a computer virus in a closed network is modelled using 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 initial detection. Determine the inverse function $t(N)$. What meaning does the domain of this inverse function have in the context of the limit of propagation of the virus?

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 whether the graph of $f(x)$ intersects the graph of its inverse function.

Given the function $f(x) = \sqrt{2x + 15}$, defined for $x \in [-7,5, +\infty)$, the goal is to determine the intersection points between its graph and that of its inverse function $f^{-1}$. Two students propose different approaches:

• Method 1 (Direct solving): Calculate the expression of $f^{-1}(x)$, set up the equation $f(x) = f^{-1}(x)$, and solve it by squaring.

• Method 2 (Symmetry): Use the property that, if a strictly increasing function intersects its inverse, the points of intersection must necessarily lie on the angle bisector line $y = x$, directly solving the equation $f(x) = x$.

1. a) Compare both methods and justify why Method 1 is significantly more complex and inefficient (analyze the degree of the resulting polynomial and the risk of extraneous roots compared to Method 2).
2. b) Apply Method 2 to solve the equation, check with justification the validity of the solutions obtained within the domain of both functions, and give the exact coordinates $(x, y)$ of the intersection point or points.
3. c) Verify that the found point indeed belongs to the graph of $f^{-1}$.

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 $(f^{-1} \circ f)(x) = f^{-1}(7)$.