Inverse functions

The idea of inverse functions is very natural when thought about in the context of functions. Although your students may not have experience with the definition of an inverse function, they do have a lot of experience with the idea of inverses – tapping into this this intuitive knowledge (like “division is the opposite of multiplication“) can help make the connections to the more abstract definition of inverse.

If a function “does” something (transforms an input into an output), then the inverse function simply “undoes” that something (transforms the given output back into the original input).

A potential issue arises if several different inputs to a function lead to the same output — in this case, it’s not clear what the inverse function should do when presented with the given output (which of the inputs should it choose?). This leads to the following restriction — for a function to have an inverse, it must be one-to-one (each output comes from a unique input or, put another way, different inputs must give different outputs).

The inverse of a function

Definition. A function $f(x)$ is one-to-one if every whenever $a$ and $b$ are in the domain of $f$ with $a\neq b$, we have $f(a)\neq f(b)$.

Definition. If $f(x)$ is a one-to-one function with domain $A$ and range $B$, then $g(x)$ is the inverse of $f(x)$ provided $g: B\rightarrow A$ and, whenever $b=f(a)$, we have $a=g(b)$.

NOTE: We often refer to the inverse function of $f(x)$ by the notation $f^{-1}(x)$, rather than $g(x)$.

Question. If $g(x)$ is the inverse function of $f(x)$ and $a$ is in the domain of $f(x)$, then what is $g(f(a))$?

Example. For each function below, find $f(5)$. If $g(x)$ is the inverse function of $f(x)$, what value $x=a$ can be plugged into $g(x)$ to obtain $g(a)=5$?

  1. The “add three” function, $f(x)=x+3$.
  2. The “subtract eleven” function, $f(x)=x-11$.
  3. The “multiply by four” function, $f(x)=4x$.
  4. The “divide by ten” function, $f(x)=\frac{x}{10}$.
  5. The “cube and add six” function, $f(x)=x^3+6$.

Algebraically, we can find a formula for the inverse function by reversing the roles of the input (x) and output (y), and then solving for y.

Example. For each function $y=f(x)$ in the example above, find a formula for the inverse function $f^{-1}(x)$. Give the inverse function a name that describes what it does to $x$ (like the names in quotes in the example).

The inverse of an exponential function

Question. What is the geometric definition of one-to-one? That is, how can you tell a function is one-to-one by looking at the graph?

Example. Consider the function $f(x)=2^x$.

  • Make a table showing the values of $f(x)$ when $x=1$, $x=4$, $x=1/2$, $x=-3$, $x=\pi$
  • What is the domain of the function $f(x)$? What is the range?
  • What does the graph look like?
  • Is $f(x)$ one-to-one?

Example. Suppose $f^{-1}(x)$ is the inverse function of $f(x)=2^x$. Find the following values of $f^{-1}$
NOTE: If you can’t give an exact answer, use your calculator (arithmetic only – you can’t use the logarithm button) to find an approximation accurate to 3 decimal places.

  1. $f^{-1}(2)$
  2. $f^{-1}(16)$
  3. $f^{-1}(\sqrt{2})$
  4. $f^{-1}(2^\pi)$
  5. $f^{-1}(5)$

Example. Find a formula for the inverse function of $f(x)=2^x$.

Since $f(x)=2^x$ is one-to-one, we should be able to find the inverse. However, we don’t have a good way of solving $x=2^y$ for $y$ — we don’t have an operation that will “get $y$ down out of the exponent”. But we know such an operation exists – it’s exactly the inverse of $2^x$. Also, if you give me $x$ and ask me to find $y$ so that $x=2^y$, we can do so by successive approximations (guess, check, and repeat), as we did in the case of $x=5$ above. Thus, we can define the function $f^{-1}(x)$, even if we haven’t provided an explicit formula for the function.

Definition. If $x$ is a positive real number, then there is a unique real number $y$ satisfying $x=2^y$. We refer to this number as the logarithm of $x$ in base 2, and we write $y=\log_2 x$.

Logarithm

Definition. If $a$ is positive real number not equal to 1, and $x$ is a positive real number, then there is a unique real number $y$ satisfying $x=a^y$. We refer to this number as the logarithm of $x$ in base a, and we write $y=\log_a x$.

NOTE: We can mentally translate the expression $\log_a x$ as “what power must we raise $a$ to in order to get $x$?” This is the sense in which “the logarithm is an exponent”.

The key fact of this definition is the equivalence between the two expressions:

$$x=a^y$$

$$y=\log_a x$$

Example. Use the definition of logarithms to answer the following:

  1. Find $\log_3 9$, and $\log_4 \frac{1}{64}$
  2. If $2^5\cdot 2^6 =2048$, what is $\log_2 2048$?
  3. Show that $\log_{10} 6 + \log_{10} 7 = \log_{10} 42$
  4. If $b=7^a\cdot 7^a \cdot 7^a$, what is $\log_7 b$?
  5. Show that $\log_a \frac{c}{d} = \log_a c – \log_a d$

Properties of Logarithms

Theorem. If $a$ is positive real number not equal to 1, and $c,d$ are positive real numbers, then the function $\log_a x$ satisfies:

  • $\log_a (cd) = \log_a c + \log_a d$
  • $\log_a \frac{c}{d} = \log_a c – \log_a d$
  • $\log_a (c^x) = x\cdot\log_a c$ (for any real number $x$)
  • $a^{\log_a c} = c$
  • $\log_a (a^c)=c$

NOTE TO SELF: Add a quote from Wu about the importance of the logarithm — see pages 166–169 of Wu A&G