HPC 1.5: Inverse Functions – Review and New Things
Instructions: Each person is to complete the following questions. Use an Nspire calculator with the Navigator
hub to answer the following questions; be sure to login. Try to construct a “big picture” view of functions and
their inverses. Work with your group. 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? Use your book to locate the definition
of Inverse Relation and write it out.
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? Use your book to locate the Inverse
Reflection Principle and write it out.
a. Why is this Principle 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 from the original function/relation
whether it’s inverse 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 12 parent functions are one-to-one? Which are not? Choose two of these
parent functions that are an example of a one-to-one function and one that is not (non-example). Include an
equation, a table of values, and a graph to illustrate why each is an example or a non-example.
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.
Finding 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.
−2
a. 𝑦 = 𝑥−5
b. 𝑓(𝑥) = √𝑥 − 2
c. 𝑓(𝑥) =
𝑥−2
𝑥+2
d. 𝑓(𝑥) = 𝑥 2 − 3𝑥 + 2
e. 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 of a function and its inverse relation? Use your book to locate the Inverse
Composition Rule and write it out below. Why is this Rule the symbolic representation?
8. Using the Inverse Composition Rule Show algebraically that 𝑓(𝑥) = 3𝑥 − 2 and 𝑔(𝑥) = 𝑥+2
are inverse
3
functions, but 𝑓(𝑥) = 𝑥 2 and 𝑔(𝑥) = √𝑥, 𝑥 ≥ 0, are not.
9. Function Properties Inherited by Inverses Some properties of functions are automatically shared by the related
inverse functions (when they exist) and some 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 Some properties of functions are not necessarily shared by the
related inverse, even if the inverse exists. 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.
11. The function h(x) has a relative maximum at the point (3, -2). Does h have an inverse function? Explain
your answer in a complete sentence.
12. The function q(x) is bounded. Suppose q has an inverse function. Could the domain of q 1 be all real
numbers? Explain your answer in a complete sentence.
13. Parting Questions and Ah-Ha’s What clarifying questions do you and your group want to ask? What new insights
did you gain about functions and their inverses that you did not know before coming to class today?