Chapter 2: Functions and Graphs
Please review this lecture (from MATH 1100 class) before you begin the section 5.7 (Inverse Trigonometric functions)
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2.7 Inverse Functions
2.6 Combinations of Functions; Composite Functions
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• Find the domain of a function.• Combine functions using the algebra of functions,
specifying domains.• Form composite functions.• Determine domains for composite functions.• Write functions as compositions.
Objectives:
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Finding a Function’s Domain
If a function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of f(x) is a real number. Exclude from a function’s domain real numbers that cause division by zero and real numbers that result in a square root of a negative number.
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Example: Finding the Domain of a Function
Find the domain of the function
Because division by 0 is undefined, we must exclude from the domain the values of x that cause the denominator to equal zero.
We exclude 7 and – 7 from the domain of g.
The domain of g is
2
5( )
49x
g xx
2 49 0x 2 49x
49x
7x ( , 7) ( 7,7) (7, )
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The Algebra of Functions: Sum, Difference, Product, and Quotient of Functions
Let f and g be two functions. The sum f + g, the difference, f – g, the product fg, and the quotient
are functions whose domains are the set of all real numbers common to the domains of f and
defined as follows:
1. Sum:
2. Difference:
3. Product:
4. Quotient:
fg
( ),f gg D D
( )( ) ( ) ( )f g x f x g x ( )( ) ( ) ( )f g x f x g x
( )( ) ( ) ( )fg x f x g x ( )
( )( )
f f xx
g g x
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Example: Combining Functions
Let and Find each of the following:
a.
b. The domain of
The domain of f(x) has no restrictions.
The domain of g(x) has no restrictions.
The domain of is
( ) 5f x x 2( ) 1.g x x
( )( )f g x 2( 5) ( 1)x x 2 6x x
( )( )f g x
( )( )f g x ( , )
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The Composition of Functions
The composition of the function f with g is denoted
and is defined by the equation
The domain of the composite function is the set of all x such that
1. x is in the domain of g and
2. g(x) is in the domain of f.
f g
( )( ) ( ( ))f g x f g x
f g
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Example: Forming Composite Functions
Given and find
f g( ) 5 6f x x 2( ) 2 1,g x x x
( ( ))f g f g x 2(2 1)f x x 25(2 1) 6x x
210 5 5 6x x 210 5 1x x
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Excluding Values from the Domain of
The following values must be excluded from the
input x:
If x is not in the domain of g, it must not be in the domain of
Any x for which g(x) is not in the domain of f must not be in the domain of
( )( ) ( ( ))f g x f g x
.f g
.f g
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Example: Forming a Composite Function and Finding Its Domain
Given and
Find
4( )
2f x
x
1
( )g xx
( )( )f g x
( )( ) ( ( ))f g x f g x1
fx
41
2x
41
2
xx
x
41 2
xx
( )( ) ( ( ))f g x f g x
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Example: Forming a Composite Function and Finding Its Domain
Given and
Find the domain of
For g(x),
For
The domain of is
4( )
2f x
x
1
( )g xx
( )( )f g x
4( )( ) ,
1 2x
f g xx
0x
12
x
( )( )f g x 1 1, ,0 0,
2 2
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Example: Writing a Function as a Composition
Express h(x) as a composition of two functions:
If and then
2( ) 5h x x
( )f x x 2( ) 5,g x x ( ) ( )( )h x f g x
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• Verify inverse functions.• Find the inverse of a function.• Use the horizontal line test to determine if a function
has an inverse function.• Use the graph of a one-to-one function to graph its
inverse function.• Find the inverse of a function and graph both functions
on the same axes.
Objectives:
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Definition of the Inverse of a Function
Let f and g be two functions such that
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
The function g is the inverse of the function f and is denoted f –1 (read “f-inverse). Thus, f(f –1 (x)) = x and
f –1(f(x))=x. The domain of f is equal to the range of
f –1, and vice versa.
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Example: Verifying Inverse Functions
Show that each function is the inverse of the other:
and
verifies that f and g are inverse functions.
( ) 4 7f x x 7( )
4x
g x
7( ( ))
4x
f g x f
7
4 74
x
7 7x x
( ( )) (4 7)g f x g x 4 7 74
x 44x x
( ( )) ( ( ))f g x g f x x
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Finding the Inverse of a Function
The equation for the inverse of a function f can be found as follows:
1. Replace f(x) with y in the equation for f(x).
2. Interchange x and y.
3. Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.
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Finding the Inverse of a Function (continued)
The equation for the inverse of a function f can be found as follows:
4. If f has an inverse function, replace y in step 3 by
f –1(x). We can verify our result by showing that
f(f –1 (x)) = x and f –1 (f(x)) = x
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Example: Finding the Inverse of a Function
Find the inverse of
Step 1 Replace f(x) with y:
Step 2 Interchange x and y:
Step 3 Solve for y:
Step 4 Replace y with f –1 (x):
( ) 2 7f x x 2 7y x 2 7x y
2 7x y 7 2x y
72
xy
1 7( )
2x
f x
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The Horizontal Line Test for Inverse Functions
A function f has an inverse that is a function, f –1, if there is no horizontal line that intersects the graph of the function f at more than one point.
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Example: Applying the Horizontal Line Test
Which of the following graphs represent functions that have inverse functions?
a. b.
Graph b represents a function that has an inverse.
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
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Graphs of f and f – 1
The graph of f –1 is a reflection of the graph of f about the line y = x.
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Example: Graphing the Inverse Function
Use the graph of f to draw the graph of f –1
( )y f x 1( )y f x
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Example: Graphing the Inverse Function (continued)
We verify our solution by observing the reflection of the graph about the line y = x.
( )y f x1( )y f x
y x