Inverse Function
Ximena is traveling to Monterrey, and the weather forecast on a U.S. website says 68 °F. She knows how to convert degrees Celsius to Fahrenheit with a formula, but now she needs to go the other way: how many degrees Celsius is that? A function that "undoes" another one is called an inverse function, and it is what you are going to master on this page.
Injective function: the condition for having an inverse
To "undo" a function, each output must come from only one input. If two different inputs gave the same output, then when going back you would not know which one to return to.
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…
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.
Finding the inverse function
Find the analytical expression for the inverse function of the affine function $f(x) = 3x - 5$.
Find the analytical expression for the inverse function of the affine function $f(x) = 3x - 5$.
Determine the inverse function of the rational function $g(x) = \frac{4}{x+2}$.
Determine the inverse function of the rational function $g(x) = \frac{4}{x+2}$.
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 $C(F)$…
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.
Inverses of exponentials and logarithms
The exponential and the logarithm are the most important example of inverse functions: each one undoes the other. With base $a$ (positive and not equal to 1):
$$\log_a(a^x)=x \qquad a^{\log_a x}=x$$With base $e$ the logarithm is called the natural logarithm and is written $\ln$: $\ln(e^x)=x$ and $e^{\ln x}=x$. That is why, to solve for an $x$ that is in an exponent, you apply the logarithm with the same base to both sides.
Notice: $(0,1)$ and $(1,2)$ are on $2^x$; their coordinates swapped, $(1,0)$ and $(2,1)$, are on $\log_2 x$.
Obtain the inverse function of the exponential function $f(x) = 2^{x-3}$.
Obtain the inverse function of the exponential function $f(x) = 2^{x-3}$.
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.…
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 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…
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?
Checking an inverse using composition
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…
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 $(g \circ f)(x) = x$.
Functions that are their own inverse
Some functions "undo themselves": applying them twice takes you back to the starting point. For example, $f(x)=-x$: if you change the sign twice, you get the original number back.
To impose the condition with parameters, there are two ways: find $f^{-1}$ and equate coefficients with $f$, or set up $f(f(x))=x$. With the conditions from the graph (points it passes through) you set up a system of equations.
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…
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…
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 $c$.
Inverse of a composition
Think about getting dressed: first you put on the sock and then the shoe. To undo it, first you take off the shoe and then the sock: the order is reversed.
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…
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.
Inverse of a piecewise function
A piecewise function is inverted piece by piece. Each piece is inverted with the usual method, and its new domain is the range that piece had in $f$.
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…
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 $f$.
Symmetry about $y=x$ and swapped domain and range
Points where $f$ and $f^{-1}$ meet
Since the two graphs are symmetric about $y=x$, a point that lies on that line is its own reflection: it belongs to both graphs at once. That is why, to find where $f$ and $f^{-1}$ meet (without computing $f^{-1}$), you solve $f(x)=x$.
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…
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…
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.
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…
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 $y = x$.
Back to the beginning: the formula is $F = \frac{9}{5}C + 32$. To undo it we solve for $C$: we subtract $32$ and multiply by $\frac{5}{9}$, so $C = \frac{5}{9}(F - 32)$. With $F=68$: $C=\frac{5}{9}(36)=20$. Ximena will have 20 °C in Monterrey.
Summary
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…
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.