Transcript section 2.7

Chapter 2:
Functions and Graphs
Please review this lecture (from MATH 1100
class) before you begin the section 5.7 (Inverse
Trigonometric functions)
2.6 Combinations of Functions; Composite Functions
2.7 Inverse Functions
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Objectives:
•
•
•
•
•
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.
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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
5x
g ( x)  2
x  49
Because division by 0 is undefined, we must exclude
from the domain the values of x that cause the
denominator to equal zero.
x 2  49  0 We exclude 7 and – 7 from
the domain of g.
x 2  49
The domain of g is
x   49
(, 7) (7,7) (7, )
x  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
f
difference, f – g, the product fg, and the quotient
g
are functions whose domains are the set of all real
numbers common to the domains of f and g ( D f Dg ),
defined as follows:
( f  g )( x)  f ( x)  g ( x)
1. Sum:
2. Difference: ( f  g )( x)  f ( x)  g ( x)
( fg )( x)  f ( x) g ( x)
3. Product:
4. Quotient:
f ( x)
f
 g  ( x)  g ( x)
 
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Example: Combining Functions
Let f ( x)  x  5 and g ( x)  x  1. Find each of the
following:
a. ( f  g )( x)  ( x  5)  ( x2  1)  x 2  x  6
2
b. The domain of ( f  g )( x)
The domain of f(x) has no restrictions.
The domain of g(x) has no restrictions.
The domain of ( f  g )( x) is (, )
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The Composition of Functions
The composition of the function f with g is denoted f g
and is defined by the equation
( f g )( x)  f ( g ( x))
The domain of the composite function f g 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.
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Example: Forming Composite Functions
2
g
(
x
)

2
x
 x  1, find f g
Given f ( x)  5 x  6 and
f g  f ( g ( x))  f (2 x 2  x  1)
 5(2 x2  x  1)  6
 10 x 2  5 x  5  6
 10 x 2  5 x  1
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Excluding Values from the Domain of ( f
g )( x )  f ( g( x ))
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 f g.
Any x for which g(x) is not in the domain of f must not
be in the domain of f g.
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Example: Forming a Composite Function and Finding Its
Domain
4
1
Given f ( x) 
and g ( x) 
x2
x
Find ( f g )( x)
4
1
4 x

( f g )( x)  f ( g ( x))  f   

1
 x 1 2
2 x
x
x
( f g )( x)  f ( g ( x))  4 x
1  2x
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Example: Forming a Composite Function and Finding Its
Domain
4
1
Given f ( x) 
and g ( x) 
x2
x
Find the domain of ( f g )( x)
For g(x), x  0
4x
,
For ( f g )( x) 
1  2x
1
x
2
1  1 

The domain of ( f g )( x) is  ,     ,0 
2  2 

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 0,  
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Example: Writing a Function as a Composition
Express h(x) as a composition of two functions:
h( x )  x 2  5
If f ( x)  x and g ( x)  x 2  5, then h( x)  ( f g )( x)
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Objectives:
•
•
•
•
•
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.
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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:
x7
f ( x)  4 x  7 and
g ( x) 
4
x7
x7


f ( g ( x))  f 
  4
7  x77  x
 4 
 4 
4x  7  7 4x
g ( f ( x))  g (4 x  7) 

x
4
4
f ( g ( x))  g ( f ( x))  x verifies that f and g are inverse
functions.
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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 f ( x)  2 x  7
Step 1 Replace f(x) with y: y  2 x  7
Step 2 Interchange x and y: x  2 y  7
Step 3 Solve for y:
x  2y  7
x  7  2y
x7
y
2
Step 4 Replace y with f –1 (x):
x7
f ( x) 
2
1
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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.
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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 1 ( x)
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x
1
y  f ( x)
y  f ( x)
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