Calculus 1.5 Inverse Functions Name: ____________________________________________
Outcomes: By the end of Section 1.5 you should:
 Determine whether a function has an inverse that is a function numerically, graphically and algebraically.
 Recognize and identify one-to-one functions and sketch the graph of the inverse.
 Find the domain of the inverse, stating any restrictions “inherited” from the original function and any
restrictions needed on the domain of the original function to guarantee that the inverse is a function.
Instructions: Each person is to complete the following questions on a separate sheet of paper. Try to construct
a “big picture” view of functions and their inverses. We will come together as a large group to answer any
questions you may have and wrap things up.
1. What is the numerical characteristic of a function and its inverse relation?
a. Why is this definition the numeric representation?
b. Give an example of this definition
2. What is the graphical characteristic of a function and its inverse relation?
a. Why is this characteristic the graphic representation?
You've seen that you sort of "flip" the original function over the line y = x to get the inverse. Using this fact, you should
notice that you can also "flip over" the Vertical Line Test to get a graphical test to tell if the inverse of a function will be a
function.
3. What is the test, and what would be a “logical” name for it? Illustrate this test with a diagram: using graphs of
continuous functions (which means they are continuous over their own domains), show an example and a nonexample of a function whose inverse is and is not a function.
4. A function is called one-to-one, when every x is paired with a unique y and every y is paired with a unique x. What
does this mean graphically?
5. One-on-One Functions Which of the functions below are one-to-one? which are not? Why?
If you need to find the domain and range, look at the original function and its graph. The domain of the original function is
the set of all allowable x-values. The range of the original function is all the y-values you'll pass on the graph. To find the
domain and range of the inverse, just swap the domain and range from the original function.
Find the inverse function, if it exists.
There will be times when you are given functions that don't have inverses. Any time you come up with a "±" sign, you can
be pretty sure that the inverse isn't a function. For example, look at what happens when I try to solve for "x =" for the
original function:
My original function: y = x2 + 1
Solving for "x =": y – 1 = x2 ⟹ ±√𝑦 − 1 = 𝑥
Well, I solved for "x =", but I didn't get a UNIQUE "x =". Instead, I've shown that any given x-value will actually correspond to
two different y-values, one from the "plus" on the square root and the other from the "minus". So, what do we do? Easy!
Restrict the domain of the function so that you are guaranteed that the inverse will also be a function. I can restrict the
domain of y = x2 +1 to x ≥ 0; I could also restrict it to x ≤ 0 or x > 0, etc., as long as the original function passes the Horizontal
Line Test.
6. Domain Restrictions to Get Inverse Functions Find the inverse and determine whether the inverse is also a function;
include a sketch of each. If the inverse is not a function, restrict the domain to guarantee an inverse function; highlight
on your sketch the portion of the function you will use. Then, find the domain and range of the inverse. Don’t forget
to consider the restrictions “inherited” from the original function.
a.
b.
c.
d.
e.
−2
𝑦 = 𝑥−5
𝑓(𝑥) = √𝑥 − 2
𝑥−2
𝑓(𝑥) =
𝑥+2
𝑓(𝑥) = 𝑥 2 − 3𝑥 + 2
𝑓(𝑥) = −√4 − 𝑥 2 ; do you get the same result if you restrict the domain differently? Explore this
further, please.
7. What is the symbolic characteristic (rule) of a function and its inverse relation
8. Using the Inverse Composition Rule Show algebraically that 𝑓(𝑥) = 3𝑥 − 2 and 𝑔(𝑥) = 𝑥+2
are inverse functions, but
3
𝑓(𝑥) = 𝑥 2 and 𝑔(𝑥) = √𝑥, 𝑥 ≥ 0, are not.
9. Function Properties Inherited by Inverses There are some properties of functions that are automatically shared by
inverse functions (when they exist) and some that are not. Suppose that f has an inverse function f -1. Give an algebraic
or graphical argument (not a rigorous formal proof) to show that each of these properties of f must necessarily be
shared by f -1.
(a) f is continuous.
(b) f is one-to-one.
(c) f is odd (graphically, symmetric with respect to the origin).
(d) f is increasing.
10. Function Properties Not Inherited by Inverses There are some properties of functions that are not necessarily shared
by inverse functions, even if the inverses exist. Suppose that f has an inverse function f -1. For each of the following
properties, give an example to show that f can have the property while f -1 does not.
(a) f has a graph with a horizontal asymptote.
(b) f has domain all real numbers.
(c) f has a graph that is bounded above.
(d) f has a removable discontinuity at x = 5.