Definition 5.2.1. Function.
A mapping \(f\) from a set \(A\) called the domain to a set \(B\) called the codomain is a function if and only if for each \(a \in A\) there exists exactly one \(b \in B\) such that \(f(a)=b\text{.}\)
Section 5.2 Inverse Functions and Exponential Functions
This section presents materials that explain or enable or use the following standards
- Calculate exponential derivatives
- Analyze a function using derivatives
In order to define exponential functions and for later sections on transcendental functions we need to review function inverses and study their calculus properties.
Subsection 5.2.1 Function Inverses
Note that the definition depends on how many elements are paired. Consider \(|x|+|y|=1\text{.}\) If \(x=3/4\text{,}\) note that both \(y=1/4\) and \(y=-1/4\) work. Thus \(y\) is not a function of \(x\) because there are two outputs. This is a nice mapping which you can graph. Try it!
Definition 5.2.2. Inverse Mapping.
The inverse mapping of a mapping \(f\) from domain \(A\) to codomain \(B\) is the mapping \(f^{-1}\) from the domain \(B\) to the codomain \(A\) such that if \(f(a)=b\) then \(f^{-1}(b)=a\text{.}\)
If the inverse mapping of a function is also a function, then it is called the inverse function. For example, \(r(\theta)=\sin(\theta)\) is a function, but its inverse is not a function, because both \(r^{-1}(0)=0\) and \(r^{-1}(\pi)=0\text{.}\)
Definition 5.2.3. One-to-One Function.
A function is one-to-one if and only if \(f(a)=f(b)\) implies \(a=b\text{.}\)
Note this definition is expressed backwards. It means the function is not the same at any two places. For example \(r(\theta)=\sin(\theta)\) is not one-to-one because \(r(0)=r(\pi)\) but \(0 \ne \pi\text{,}\) that is \(r(\theta)\) has the same value at two different places.
Activity 5.2.1. Connecting Inverse and 1-1 Functions.
The goal of this activity is to recognize that the definitions of one-to-one function and inverse function match.
(a)
If \(f(1)=f(-1)=1\) is the function one-to-one?
(b)
Suppose \(f(1)=f(-1)=1\text{.}\) What would \(f^{-1}(1)\) be? Could the inverse (\(f^{-1}\)) be a function?
(c)
If the function does not repeat a value (i.e., is one-to-one), will the inverse be a function?
Activity 5.2.2. Connecting 1-1 and Increasing/Decreasing Functions.
The goal of this activity is to recognize a connection between being strictly increasing/decreasing and being one-to-one.
(a)
(b)
If \(f(x)\) is strictly increasing is it one-to-one? Why?
(c)
If \(f(x)\) is strictly decreasing is it one-to-one? Why?
(d)
What does this imply about \(\ln(x)\text{?}\)
Subsection 5.2.2 Slopes on Inverse Functions
Activity 5.2.3. Calculating Derivatives of Inverse Functions.
The goal of this activity is to recognize the relationship between slopes on a function and on its inverse.
(a)
Note \((0,1)\) and \((4,2)\) are points on \(\log_2(x)\text{.}\) What two points does this imply are on the inverse function?
(b)
What is the slope of the line containing \((0,1)\) and \((4,2)\text{?}\) What is the slope of the line containing the two points on the inverse?
(c)
Note \((0,0)\) and \((2,8)\) are points on \(f(x)=x^3\text{.}\) What two points does this imply are on \(f^{-1}(x)=x^{1/3}\text{?}\)
(d)
What is the slope of the line containing \((0,0)\) and \((2,8)\text{?}\) What is the slope of the line containing the two points on the inverse?
(e)
In general what is the relationship between the slope of a secant line on a function and the slope of the secant line on the inverse function from the matching points?
(f)
Recall that the derivative is the limit of the slopes of the secants, so this property applies to derivatives as well.
