Calculus 1.5 Inverse Functions Worksheet Name: ______KEY_________
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: Complete the following questions on a separate sheet of paper. Try to construct a “big picture” view of functions and their inverses. Write answers to all questions in complete sentences. All graphs and sketches should follow the rule of thumbs. Read all questions carefully.
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.
The ordered pair (a,b) is in a relation if and only if the ordered pair (b,a) is in the inverse relation. a.
Why is this definition the numeric representation?
This is a numeric representation because it concerns switching the coordinates of ordered pairs in a relation. In other words, it concerns switching numbers. 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.
The graph of a relation and the graph of its inverse relation are reflections of each other over the line y = x.
The inverse reflection principle states: The points (a,b) and (b,a) in the coordinate plane are symmetric with respect to the line y = x. The points (a,b) and (b,a) are reflections of each other across the line y = x.
Example: If the ordered pair (3,2) is part of a relation, then the ordered pair (2,3) must be part of the
inverse relation. a.
Why is this Principle the graphic representation?
This principle is a graphic representation because it concerns the symmetrical properties of the graph of
a relation and the graph of its inverse relation.
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.
Horizontal Line Test: The inverse of a relation is a function if and only if each horizontal line intersects the graph of the original relation in at most one point.

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?
A one-to-one function passes the horizontal line test. This means that every horizontal line drawn through a one-to-one function will intersect the graph in at most one point.
5.
One-on-One Functions Which of the 12 parent functions are one-to-one? which are not? Include an equation, a table of values, and a graph to illustrate why each is an example or a non-example.
One-to-one One-to-one One-to-one One-to-one One-to-one
Not One-to-one
One-to-one One-to-one
Not One-to-one Not One-to-one Not One-to-one Not One-to-one
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 = x 2 + 1 Solving for "x =": y – 1 = x 2 ⟹ ±√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 = x 2 +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 1 and determine whether the inverse is also a function 2 ; include a sketch 3 of each. If the inverse is not a function, restrict 4 the domain to guarantee an inverse function; highlight 5 on your sketch the portion of the function you will use. Then, find the domain 6 and range 7 of the inverse. Don’t forget to consider the restrictions “inherited” from the original function. a.
𝑦 =
−2 𝑥−5
Inverse: y

 x
2

5 The inverse is a function.

Calculus 1.5 Inverse Functions Worksheet Name: ______KEY_________
Domain of the inverse: (

, 0)

(0, )
Range of the inverse: (

, 5)

(5, ) b.
𝑓(𝑥) = √𝑥 − 2
Inverse: y
 x
2 
2 The inverse is a function.
Domain of the inverse: [0, )
Range of the inverse: c.
𝑓(𝑥) = 𝑥−2 𝑥+2

Inverse: y

 
2 x

1
The inverse is a function.
Domain of the inverse: ( ,1) (1, )
Range of the inverse: ( , 2) ( 2, ) d.
𝑓(𝑥) = 𝑥 2 − 3𝑥 + 2
Inverse: y
3
2 x

1
4
The inverse is not a function.
Domain restriction on original function:
3
[ , )
2

Domain of the inverse: [

1
4
, )

Calculus 1.5 Inverse Functions Worksheet
Range of the inverse:
3
[ , )
2

Name: ______KEY_________ e.
𝑓(𝑥) = −√4 − 𝑥 2 ; do you get the same result if you restrict the domain differently? Explore this further, please.
Inverse: y
  x
2 
4 The inverse is not a function
Domain restriction on the original function:
 
Domain of inverse:
 
2, 0

Range of Inverse:
 
If you restrict the domain of the original to
 
2, 0

Then the domain of the inverse is of the inverse is
 
2, 0

.
 
2, 0
 and the range
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?
Inverse Composition Rule: A function f is one-to-one with inverse function g if and only if f(g(x)) = x for every x in the domain of g, and g(f(x)) = x for every x in the domain of f.
This is a symbolic representation because it concerns equations rather than graphs or ordered pairs.
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.

( ( ))
 f ( x

2
)

3 x

2
2
 x

2

2
3 3
( ( ))
 g

3 x

2
  3 x

3
2
 
2

3
3 x
 x inverses
( ( ))
 f ( x )

( ( ))

 
 x
Therefore 𝑓(𝑥) = 3𝑥 − 2 and 𝑔(𝑥) = 𝑥+2
are
3
2
 x x
2  x
 x
Therefore 𝑓(𝑥) = 𝑥 2
and 𝑔(𝑥) = √𝑥 are not inverses
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.
If the graph of f is unbroken, its reflection over the line y = x will also be unbroken.
(b) f is one-to-one.
Both f and its inverse must be one-to-one in order to be inverse functions.
(c) f is odd (graphically, symmetric with respect to the origin).
Since f is odd, (-x,-y) is on the graph whenever (x,y) is. This implies that (-y,-x) is on the graph of the inverse whenever (y, x). Therefore the inverse must also be odd.
(d) f is increasing.
If the original function f is increasing then this means that a positive change in x results in a positive change in y. For the inverse this means that a positive change in y (the input for the inverse) also results in a positive change in x (the output for the inverse). Therefore the inverse must also be 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.
Example: ( )
 e x has a horizontal asymptote. f

1
( )
 x has a vertical asymptote.
(b) f has domain all real numbers.
( )
 e x has domain

,

. f

1
( )
 ln( ) has domain

0,
 
.
(c) f has a graph that is bounded above.
( )
 e x is bounded below. f

1 x
 x is not bounded.
(d) f has a removable discontinuity at x = 5.

 x

5
 x

5
 x

5
has a removable discontinuity at x = 5. The graph of f is the line y = x + 5 with the point (5, 10) removed. The inverse function is the line y = x – 5 with the point (10, 5) removed. This function has a removable discontinuity, but not at x = 5.