7 Inverse Functions
TERMINOLOGY
Domain: Set of all possible x Domain: Set possible x values values (the independent variable) of a function
Mutually inverse functions: Two functions: Two functions are mutually inverse if f 1[f ( x x )] )] f [f 1( x x )] )] x
Horizontal line test: Used test: Used to test for the existence of inverse functions. A function’s inverse relation is a function if any horizontal line cuts the original function in one place only
Range: Set of all possible y possible y values values (the dependent variable) of a function
Inverse function: An function: An inverse function is the opposite of (or undoes) the original function. The domain becomes the range and the range becomes the domain. It is a result of reflection of the original function in the line y line y x =
-
=
-
=
Restricted domain: Domain domain: Domain restricted to the x the x values values that will make the inverse relation a function
Chapter 7 Inverse Functions
INTRODUCTION IN THIS CHAPTER YOU will study inverse functions. functions.
In particular, you will learn about inverse trigonometric functions, functions, including differentiation and integration integration of of these functions.
Inverse Functions When solving or changing the subject of an equation, we use inverse operations. For example, to solve x + 3 = 7, we subtract 3 from both sides. Inverse functions are formed by taking the inverse operation or operation or ‘undoing ’ the operation of the function. However, However, the inverse is not always a function.
DID YOU KNOW? The notation for inverse functions was first used by the astronomer Sir William Herschel (1738–1822). Born in Germany,, Herschel discovered Uranus, two of its satellites, Germany and two satellites of Saturn. He also discovered infrared radiation. His sister and his son were also astronomers.
From left to right: Kepler, Kepler, Herschel and Newton on the Astronomers Monument at the Griffith Observatory in Los Angeles
335
Chapter 7 Inverse Functions
INTRODUCTION IN THIS CHAPTER YOU will study inverse functions. functions.
In particular, you will learn about inverse trigonometric functions, functions, including differentiation and integration integration of of these functions.
Inverse Functions When solving or changing the subject of an equation, we use inverse operations. For example, to solve x + 3 = 7, we subtract 3 from both sides. Inverse functions are formed by taking the inverse operation or operation or ‘undoing ’ the operation of the function. However, However, the inverse is not always a function.
DID YOU KNOW? The notation for inverse functions was first used by the astronomer Sir William Herschel (1738–1822). Born in Germany,, Herschel discovered Uranus, two of its satellites, Germany and two satellites of Saturn. He also discovered infrared radiation. His sister and his son were also astronomers.
From left to right: Kepler, Kepler, Herschel and Newton on the Astronomers Monument at the Griffith Observatory in Los Angeles
335
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Maths In Focus Mathematics Extension 1 HSC Course
EXAMPLES The inverse of multiplying by 2 is dividing by 2.
1. The inverse relation of y
=
2. The inverse relation of y
=
3. The inverse relation of y
=
2x is y
=
x is y e
x
is y
=
x 2
. 2
=
x .
loge x.
7.1 Exercises Write down the inverse relation of each of the following functions. 1.
y
=
3x
11.
y
=
log4 x
2.
y
=
−
x
12.
y
=
x
3.
f (x)
4.
y
=
5.
y
=
6.
f (x) = x + 1
7.
y
8.
f (x) = x + 3
9.
y
3
=
=
x 5
=
x
7x
x
x
10. f (x)
−
5
3
=
2x
5
13. f (x)
=
x
−
9
14. f (x)
=
5
−
x
15.
y
=
−
16.
y
=
x
17.
y
=
18.
y
=
19.
y
=
20.
y
=
3x
2
7
x
loge
x
x 9
x
8
It is harder to find the inverse relation when more than one operation is involved.
EXAMPLES Find the inverse relation of each of the following: 1.
y
=
2x
+
1
Solution Changing the subject of the equation to x by using inverse operations will show what these operations are. y
−
y
=
2x
1
=
2x
=
x
+
1
y − 1 2
Both y = 2x + 1 and
y x
=
−
2
1
represent the same relation.
Chapter 7 Inverse Functions
The inverse operations of ‘multiplying by 2 then adding 1’ are ‘subtracting 1 then dividing by 2’. This means that the inverse relation of y = 2x + 1 is y
x
=
1
−
.
2
Changing the subject of this inverse relation gives x = 2y + 2.
y
x
=
3 −
1.
2
Solution Changing the subject of the equation to x by using inverse operations will show what these operations are.
3
y
=
x
y
+
2
=
x
y
+
2
=
x
3
−
2
3
Both y x 3 2 and x = 3 y + 2 represent the same relation. The inverse operations of ‘cubing then subtracting 2’ are ‘adding 2 then finding the cube root’. This means that the inverse relation of y x 3 2 is y = 3 x + 2 . Changing the subject of this inverse relation gives x y 3 2. =
−
=
−
=
The inverse relation of y values of values of the function.
=
f (x) can be found by interchanging the x the x and y
EXAMPLES 1. Find the inverse relation of y
=
3x
−
8.
Solution x
=
3y − 8
=
3y
x
+
8
x
+
8 = y
3
2. Find the inverse relation of f (x) = 2x 5 + 7.
Solution x x
−
7
x
−
7
x
−
2
=
5
2y
5
2y 5
= y
2 5
=
7 = y
+
7
−
Can you find an easy way to find the inverse relation?
337 33 7
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Maths In Focus Mathematics Extension 1 HSC Course
Sometimes the inverse relation is harder to find.
EXAMPLES Find each inverse relation 1.
=
y
x
2
3
+
Solution 2
=
x
+
3
4x
−
y
2
x–3
=
y
3
=
y
±
x
2.
y
−
=
x
2
+
7
Solution x
Complete the square on y
2
+
4y
+
2 2
+
4y – 7
+
4y
+
4y + 4
=
y
7+4
=
y
=
(y + 2) 2
=
y +
=
y
11
+
x Notice that in these examples, the inverse is not a function.
y
7
+
x x
=
± x + 11 ± x + 11
−
2
2
2
7.2 Exercises Find the inverse relation of each of the following functions 1.
y
=
5x
2.
y
=
2x
3.
y
=
x
4.
f (x)
5.
y
=
6.
y
=
7.
y =
8.
y =
9.
f (x) =
10.
y
=
11.
y
=
3
13.
5
2
−
5
15.
y
=
16.
y
=
e
17.
y
=
e
18.
y
=
ln x
19.
y
=
ln (x
3
x
3x
14. f (x) = 2 x
x7
=
=
3
−
+
y
−
−
1
2
2
x 3
3
3
2x
+
+
1
x
2x
+
1)
x
+
5
x
+
1
20. f (x) = e 3x + 1
x+2
21.
y
=
7
22.
y
=
23.
y
=
x
24.
y
=
x –3
2
3
x
−
3
x
2
2x 2
4
+
5
x
12. f (x) = 5x 3 + 1
6
5
Chapter 7 Inverse Functions
25.
y
=
x2
26.
y
=
4x
27.
y
=
x2
+
8x
28.
y
=
x2
+
10x
x2
29.
y
=
x2
−
6x
30.
y
=
x2
+
12x
−
−
2x
+
3
−
1
3
−
−
11
Graph of Inverse Functions Class Investigation Sketch pairs of functions with their inverse relations on the same number plane. What do you notice about their graphs?
On the number plane, the inverse relation can be represented by a reflection of the original function in the line y x For example, y x 2 gives x y 2 when reflected in the line y x Notice that, x y 2 is not a function. =
=
.
=
=
.
=
y
x2
5
y y
x
5
x
x
y2
5
Class Investigation Find the sketch of the inverse relation of a function by reflecting it in the line y x =
.
EXAMPLE
y
=
x
3
Use thin (e.g. tracing) paper and fold along the line y inverse relation.
=
x to
see the
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340
Maths In Focus Mathematics Extension 1 HSC Course
Graph the inverse relation of each function below. Function Inverse y
=
x
y
=
x
y
=
e
y
=
x
y =
2
+
+
1
3
x 2
+
x
−
2
+
1
=
y
x
=
y + 3
x
=
e
x
=
y
x
=
1
x
2
x
y 2
+
y − 2
1
y
• Which of these inverse relations are functions? • How could you test the original function to see if its inverse is a function?
Horizontal line test A function has a unique y value for every x value. Since the inverse relation is an exchange of the x and y values, the inverse is a function if there is a unique value of x for every y value in the original function. This means that the original function must be a one-to-one relation— that is, there is a unique x value for every y value, and a unique y value for every x value. This occurs when a graph is monotonic increasing or decreasing. In the Preliminary Course, you used a vertical line test to check a function. To check if the inverse relation is a function, we use a horizontal line test.
EXAMPLES Are the inverse relations of the following curves functions? 1.
y
=
x
3
A horizontal line cuts the curve in only one place. Thus the inverse will be a function.
Chapter 7 Inverse Functions
2.
y
=
x
2
A horizontal line can cut the curve more than once, so the inverse is not a function.
7.3 Exercises Do these functions have an inverse function? 1.
3.
2.
4.
341
342
Maths In Focus Mathematics Extension 1 HSC Course
5.
9.
y
x
6.
10.
y
y
x
7.
x
11.
y
y
x
x
8.
12.
y
y
x x
