Warmup Given the following functionsstaffweb.psdschools.org/kemotich/Mrs_Motichka/Precalculus...4.2...
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4.2 OnetoOne Functions and Inverse Functions 2011 1 April 20, 2011 Warmup Given the following functions: 1. 2.
Transcript of Warmup Given the following functionsstaffweb.psdschools.org/kemotich/Mrs_Motichka/Precalculus...4.2...
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Warmup
Given the following functions:
1. 2.
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4.2 OnetoOne Functions; Inverse Functions
Objectives: • Determine whether a function is onetoone.• Obtain the graph of the inverse function from the graph of the function.• Find the inverse of a function defined by an equation.
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On the graph, sketch an example of a function.
Why do you know that it is a function?
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A function is onetoone if each input has its own output.
How to check:• Every x value has its own y value.• The graph passes the horizontal line test.
Examples:#1 (0, 0), (1, 1), (2, 16), (3, 81)
#2
Functions can be categorized as onetoone or manytoone.
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Onetoone Manytoone Not a function
1010
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EXAMPLES
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Mapping FunctionsOnetoone Manytoone Not a function
each x has only one y is the image of two x's is paired with two y's
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Determine whether each given function is onetoone:
1. Domain Range 2. (1, 4), (2, 5), (3, 6), (4, 6) Jeffrey Liz Benjamin Ben Carolyn Carol Elizabeth Jeff
3. 4.
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If a function f is onetoone, then it has an inverse function f 1.
The graphs of f and f 1 are symmetric with respect to the line y = x.
Example:
f(x)
f 1(x)
Domain of f = Range of f 1
Range of f = Domain of f 1
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x y
Graph the inverse function:
10123
10123
x y53113
53113
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Graph the inverse function:
x y10123
53113
x y10123
53113
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The graph of a onetoone function is given.Draw the graph of the inverse function f 1.For convenience, the graph of y = x is also given.
(4, 2)
(3, 0)(0, 1)
(2, 5)
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To verify that f 1 is the inverse of f, show 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.
Given: f(x) = 4x 8 f 1(x) = x/4 + 2
Show:f 1(f(x)) = x f(f 1(x)) = x
f 1(4x 8) = x f(x/4 + 2) = x
4x 8 4(x/4 + 2) 8 = x 4
x + 8 8 = xx 2 + 2 = x
x = xx = x
+ 2 = x
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Finding the Inverse Function:
Given: f(x) = x ≠ 1
Step 1: Interchange the variables x and y.
Step 2: Solve for y.
Step 3: Rewrite as f 1(x)
2x + 1x 1
2x + 1x 1
y = x = 2y + 1y 1
x = 2y + 1y 1
(y 1)(y 1)
xy x = 2y + 1 xy 2y = x + 1y(x 2) = x + 1
y = x + 1x 2
x + 1x 2
f 1(x) =
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Step 4: Check the result showing that f 1(f(x)) = x and f(f 1(x)) = x
f(x) = x ≠ 12x + 1x 1
x + 1x 2
f 1(x) =
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Step 5: Find the domain and range of f and f 1. HINT: Remember Domain of f = Range of f 1 and Domain of f 1 = Range of f.
f(x) = x ≠ 12x + 1x 1
x + 1x 2
f 1(x) =
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HOMEWORK: page 267 (10, 12, 14, 16, 19 22, 32, 34, 37, 42, 43, 45, 48 58 even & no graphing, 76)
