one-to-one functions; inverse function. a function f is one-to-one if for each x in the domain of f...
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One-to-One Functions; Inverse Function
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A function f is one-to-one if for each x in the domain of f there is exactly one y in the range and no y in the range is the image of more than one x in the domain.
A function is not one-to-one if two different elements in the domain correspond to the same element in the range.
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x1
x2
x3
y1
y2
y3
x1
x3
y1
y2
y3
x1
x2
x3
y1
y3Domain
Domain
Domain
Range
Range
RangeOne-to-onefunction
Not a function
NOT One-to-onefunction
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M: Mother Function is NOT one-one
Joe
Samantha
Anna
Ian
Chelsea
George
Laura
Julie
Hilary
Barbara
Sue
Humans Mothers
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S: Social Security function IS one-one
Joe
Samantha
Anna
Ian
Chelsea
George
123456789
223456789
333456789
433456789
533456789
633456789
Americans SSN
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Is the function f below one – one?
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10
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16
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Theorem Horizontal Line Test
If horizontal lines intersect the graph of a function f in at most one point, then f is one-to-one.
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Use the graph to determine whether the function f x x x( ) 2 5 12
is one-to-one.
Not one-to-one.
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Use the graph to determine whether the function is one-to-one.
One-to-one.
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The inverse of a one-one function is obtained by switching the role of x
and y
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123456789
223456789
333456789
433456789
533456789
633456789
SSN
Joe
Samantha
Anna
Ian
Chelsea
George
Americans
The inverse of the social security function1S
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Let and
Find
3)( xxf 31
)( xxg
))(())(( xfgandxgf
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g is the inverse of f.
311 )( xxf
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.Let f denote a one-to-one function y = f(x). The inverse of f, denoted by f -1 , is a function such that f -1(f( x )) = x for every x in the domain of f and f(f -1(x))=x for every x in the domain of f -1.
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Domain of f Range of f
Range of f 1 Domain of f 1
f 1
f
Domain of Range of
Range of Domain of
f f
f f
1
1
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Theorem
The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.
f 1
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2 0 2 4 6
2
2
4
6 f
f 1
y = x
(2, 0)
(0, 2)
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Finding the inverse of a 1-1 function
Step1: Write the equation in the form
Step2: Interchange x and y.
Step 3: Solve for y.
Step 4: Write for y.
)(xfy
)(1 xf
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Find the inverse of
Step1:
Step2: Interchange x and y
Step 3: Solve for y
3
5
xy
3
5)(
xxf
3
5
yx
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x
xxf
35)(1
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