Chapter 3: The Nature of Graphs
Section 3-4: Inverse Functions and Relations
Inverse Relations
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Two relations are inverse relations if and only if one
relation contains the element (b, a) whenever the other
relation contains the element (a, b).
-1
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Notation: If f (x) denotes a function, then f ( x)
denotes the inverse of f (x). But remember the inverse
might not be a function.
You can use the horizontal line test to determine if the
inverse of a relation will be a function. If every horizontal
line intersects the graph of the relation in at most one
point, then the inverse of the relation is a function.
Example # 1
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2
Consider f ( x) = x − 4
Is the inverse of f (x) a function NO
Find f −1 ( x ) and graph both f (x) and its inverse.
Solution:
We know that the function f (x) is a parabola child that looks like mom and
is translated 4 units down.
To find the inverse, I switch the x and y in f (x) and solve for y.
x = y2 − 4
The y in the inverse is NOT the same y as in f (x)
x + 4 = y2
± x + 4 = y = f −1 ( x)
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Notice: The graph of the inverse is
a reflection of the original graph
over the line y=x
Inverse Functions
Two functions,
f and
if and only if ⎡⎢f o
⎣
Example: Given
Solution:
f -1
Are inverse functions
⎡
⎤
f -1⎥ ( x) = ⎢f -1 o
⎣
⎦
⎤
f ⎥ ( x) = x
⎦
f ( x) = 3 x 2 + 7,
find f −1 ( x),
and verify that f and f -1
are inverse functions
f −1( x) =
x−7
;
3
⎡
−1 ⎤
⎢ f o f ⎥ ( x) =
⎣
⎦
⎛
⎜
3⎜
⎜
⎝
⎛
⎜
f⎜
⎜
⎝
2
⎞
x−7 ⎟
⎟ +7= x
3 ⎟
⎠
⎞
x−7 ⎟
⎟=
3 ⎟
⎠
and
⎡ −1
⎢f o
⎣
⎤
f ⎥ ( x) = f −1(3 x 2 + 7) =
⎦
(3 x 2 + 7) − 7
=x
3
⎡
⎤
⎡
So ⎢ f o f −1 ⎥ ( x) = ⎢ f −1 o
⎣
⎦
⎣
⎤
f ⎥ ( x) = x
⎦
HW # 19
Section 3-4
„ Pp. 156-158
„ #15,16,17,22,25,28,33,46
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