Section 4.1: Inverse Functions.
Vertical Line Test: A function takes every value in its domain and assigns to it a unique value in its range. Graph-
ically, this means that it passes the vertical line test: Any vertical line can intersect the graph of a function at
most once.
In the four graphs illustrated below, graph B does not represent a function because it fails the vertical line test: The
line x= 1 intersects the graph four times.
The Horizontal Line Test: Notice that it is possible for functions to have multiple domain values evaluate to the
same range value, as illustrated in graphs A and C. They both have multiple x-intercepts evaluating to the same
value: y= 0.
When a function has every range value corresponding to exactly just one domain value, it is said to be one-to-one
or invertible. Graphically, this means that it passes the horizontal line test: Any horizontal line can intersect the
graph of a one-to-one function at most once.
In our four graphs above, only graph D is a one-to-one function.
Inverse Functions: If a function f(x)is one-to-one, then the inverse function of f(x)is denoted as f−1(x)and is
read as “finverse of x”. It has the following properties:
1. Algebraically, their mutual composition yields x. That is, (f◦f−1)(x)=(f−1◦f)(x)=x.
2. Graphically, the graph of f(x)and f−1(x)will be symmetric along the line y=x.
3. If the point (x, y)is on the graph of f(x), then the point (y, x)will be on graph of f−1(x).
Just like section 2.5, you will need to be able to do the “algebraic method” of moving points/intervals around.
Example: The point (a, b)is on the graph of the one-to-one function y=f(x). For each of the following functions,
enter the ordered pair that corresponds to the transformation of (a, b).
(a) New Function: y= 4f(x−7) + 9
(b) New Function: y= 5f−1(6x−3) −2
(c) New Function: y=−3f−1(5 −x)+4
And just like section 2.7, you will be tasked with the same three things: computing the inverse function, evaluating
the inverse algebraically or by using a chart, and analyzing domains of inverse functions.
Task #1: Computing the Inverse:
We will be using the third property listed above to compute inverses. Since inverses swap the xand ycoordinates,
we will do the same to compute f−1(x). To compute the inverse of a function, you swap your xand yvariables,
then solve for y.
Examples:
1. Compute the inverse function for f(x)=x3+ 7.
2. Compute the inverse function for g(x)=x2−27,x≤0.
