Math 1314
2.7 Inverse Functions
Section 2.7 Notes
Review:
Vertical line test:
If every vertical line crosses the graph of a relation at most once, then the graph defines a function.
Example: For each graph below. Use the vertical line test to determine whether it defines a function.
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Math 1314
Section 2.7 Notes
One-to-One Functions
Definition: A function f is said a one-to-one function (1-1 function) if each value for f(x) corresponds to only
one value for x.
Remark: f is a one-to-one function if f ( x1 )  f ( x2 ) implies x1  x2 .
Examples:
1. Function f(x) = 2x + 1 is a one-to-one function because
Suppose f ( x1 )  f ( x2 ) then we have
f ( x1 )  f ( x2 )  2 x1  1  2 x2  1
 2 x1  2 x2
 x1  x2
2. Function f ( x )  x 2 is not a one-to-one function because
Suppose f ( x1 )  f ( x2 ) then we have
f ( x1 )  f ( x2 )  x1  x2
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2
 x1  x2 or x1   x2
Horizontal line test:
If every horizontal line crosses the graph of a function at most once, then that function is one-to-one.
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Example: Use the graphing calculator to graph the function f ( x )  x then use the horizontal line test to
determine whether this function is one-to-one.
This is not a one-to-one function because it fails the horizontal line test.
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Section 2.7 Notes
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Example: Use the graphing calculator to graph the function f ( x)  x then use the horizontal line test to
determine whether this function is one-to-one.
This is a one-to-one function.
Example: Use the graphing calculator to graph the function f ( x )  x 2 then use the vertical line test to determine
whether this function is one-to-one.
2.7 Inverse Functions
What does INVERSE mean?
* Addition and Subtraction are inverse operations of each other. x  7  7  x
* Multiplication and Division are inverse operations of each other. x  7  7  x
The inverse operations UNDO each other.
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Definition:
Section 2.7 Notes
If f is a one-to-one function, then g is the inverse function of function f if:

f(g(x)) = x for all x in the domain of g.

g(f(x)) = x for all x in the domain of f.
1
Notation: If function f(x) has an inverse function, we write f ( x ) to denote the inverse function for f(x).
then f 1  f ( x)   x and f  f 1 ( x)   x for all x’s in the domain of f.
Let D denote the domain, R denote the range, then D f  R f 1 and D f 1  R f
Note: f 1  x  does not indicate the exponent that means f 1 x  
1
.
f ( x)
Critical thinking: A function f has an inverse function if and only if f is a one-to-one function. WHY?
To verify two functions are inverses:
Functions f(x) and g(x) are inverses of each other if:

( f  g )( x)  x for all x in the domain of g.

( g  f )( x)  x for all x in the domain of f.
Example: For each pair of functions, determine whether they are inverses of each other.
1.
f ( x )  4 x  9 and g ( x ) 
x 9
4
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Section 2.7 Notes
x3
7
2.
f ( x )  3x  7 and g ( x ) 
3.
f ( x)  3 x  4 and g ( x)  x 3  4
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Math 1314
Steps for finding the inverse function of a function f:
Section 2.7 Notes
1. Verify that f is one-to-one function.
2. Replace f(x) with y.
3. Interchange x and y.
4. Solve for y.
5. Replace y with g(x).
6. Check whether f(x) and g(x) are inverses functions of each other. If so replace g ( x) with f 1 x  .
If not, we may need to restrict the domain of g(x) to get the inverse function of f(x).
Example: For each one to one function, find its inverse function.
1.
f ( x)  2 x  3
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2.
f ( x) 
Section 2.7 Notes
7
3
x
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3.
Section 2.7 Notes
f ( x)  x3  2
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4.
Section 2.7 Notes
f ( x)  x
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5.
Section 2.7 Notes
f ( x )  x 2  3 where x  0
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6.
f ( x) 
Section 2.7 Notes
2x  1
x 3
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Section 2.7 Notes
Graphs of f ( x ) and f 1 ( x )
The graph of f 1 ( x ) is a reflection of the graph of f (x) about the line y = x.
Graphing the inverse function
Example: Given graph of one to one function f, sketch the graph of f -1.
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2.
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Section 2.7 Notes
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