inverses algebraically 2 objectives i can find the inverse of a relation algebraically
TRANSCRIPT
Inverses Algebraically
2
Objectives
• I can find the inverse of a relation algebraically
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NOTATION FOR THE INVERSE FUNCTION
f x 1 ( )
f x 1 ( )
We use the notation
for the inverse of f(x).
NOTE: does NOT mean 1
f x( )
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Inverses
• The inverse of any relation is obtained by switching the coordinates in each ordered pair of the relation.
• Example:
• { (1, 2), (3, -1), (5, 4)} is a relation
• { (2, 1), (-1, 3), (4, 5) is the inverse.
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The ordered pairs of the function f are reversed to produce the ordered pairs of the inverse relation.
Example: Given the function f = {(1, 1), (2, 3), (3, 1), (4, 2)}, its domain is {1, 2, 3, 4} and its range is {1, 2, 3}.
The inverse relation of f is {(1, 1), (3, 2), (1, 3), (2, 4)}.
The domain of the inverse relation is the range of the original function.
The range of the inverse relation is the domain of the original function.
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DomainRange
Inverse relation x = |y| + 1
210-1-2
x y
321
Domain Range
210-1-2
x y
321
Function y = |x| + 1
Every function y = f (x) has an inverse relation x = f (y).
The ordered pairs of : y = |x| + 1 are {(-2, 3), (-1, 2), (0, 1), (1, 2), (2, 3)}.
x = |y| + 1 are {(3, -2), (2, -1), (1, 0), (2, 1), (3, 2)}.
The inverse relation is not a function. It pairs 2 to both -1 and +1.
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FINDING A FORMULA FORAN INVERSE FUNCTION
To find a formula for the inverse given an equation for a one-to-one function:
1. Replace f (x) with y.
2. Interchange x and y.
3. Solve the resulting equation for y.
4. Replace y with f -1(x) if the inverse is a function.
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Example: Find the inverse relation algebraically for the function f (x) = 3x + 2.
y = 3x + 2 Original equation defining f
x = 3y + 2 Switch x and y.
3y + 2 = x Reverse sides of the equation.
y = Solve for y.3
)2( x
To find the inverse of a relation algebraically, interchange x and y and solve for y.
1 1 2( )
3 3f x x
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Example 2: Find the inverse relation algebraically for the function f (x) = x2 + 6.
y = x2 + 6 Original equation defining f
x = y2 + 6 Switch x and y.
y2 = x - 6 Reverse sides and subtract 6
y = Solve for y.6x
Is this a function??
Find the inverse function “algebraically”
f(x) = 6
Exchange the x & y values
x = 6Is the inverse a function?
NO, because it is a vertical line. You can find the inverse but not the inverse function.
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TESTING FOR AONE-TO-ONE FUNCTION
Horizontal Line Test: A function is one-to-one (and has an inverse function) if and only if no horizontal line touches its graph more than once.
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x
y
2
2
Horizontal Line Test
A function y = f (x) is one-to-one if and only if no horizontal line intersects the graph of y = f (x) in more than one point.
y = 7
Example: The function
y = x2 – 4x + 7 is not one-to-one
on the real numbers because the
line y = 7 intersects the graph at
both (0, 7) and (4, 7).
(0, 7) (4, 7)
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one-to-one
Example: Apply the horizontal line test to the graphs below to determine if the functions are one-to-one.
a) y = x3 b) y = x3 + 3x2 – x – 1
not one-to-one
x
y
-4 4
4
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x
y
-4 4
4
8
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y = f(x)y = x
y = f -1(x)
Example: From the graph of the function y = f (x), determine if the inverse relation is a function and, if it is, sketch its graph.
The graph of f passes the horizontal line test.
The inverse relation is a function.
Reflect the graph of f in the line y = x to produce the graph of f -1.
x
y
Homework
• WS 3-1
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