bellwork 1-15-15 y + 5 8 2. x = 4. x = yedweb.tusd1.org/jdumes/documents/algebra 2/4-2...
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42 Inverses of Relations and Functions.notebook
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Bellwork 1-15-15Solve for y.
1. x = 3y –7
4. x = y2
3. x = 4 – y
2. x = y + 58
42 Inverses of Relations and Functions.notebook
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Graph and recognize inverses of relations and functions.
Find inverses of functions.
Objectives
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inverse relationinverse function
Vocabulary
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You have seen the word inverse used in various ways.The additive inverse of 3 is –3.
The multiplicative inverse of 5 is
The multiplicative inverse matrix of
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You can also find and apply inverses to relations and functions. To graph the inverse relation, you can reflect each point across the line y = x. This is equivalent to switching the x- and y-values in each ordered pair of the relation.
A relation is a set of ordered pairs. A function is a relation in which each x-value has, at most, one y-value paired with it.
Remember!
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Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation.
Example 1: Graphing Inverse Relations
x 0 1 5 8
y 2 5 6 9
Graph each ordered pair and connect them.
x 2 5 6 9y 0 1 5 8
Switch the x- and y-values in each ordered pair.
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Example 1 Continued
Reflect each point across y = x, and connect them. Make sure the points match those in the table.
Domain:{x|0 ≤ x ≤ 8}Range :{y|2 ≤ x ≤ 9}
Domain:{x|2 ≤ x ≤ 9}Range :{y|0 ≤ x ≤ 8}
x 2 5 6 9y 0 1 5 8
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Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation.
Check It Out! Example 1
Graph each ordered pair and connect them.
Switch the x- and y-values in each ordered pair.
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Reflect each point across y = x, and connect them. Make sure the points match those in the table.
Domain:{1 ≤ x ≤ 6}Range :{0 ≤ y ≤ 5}
Domain:{0 ≤ y ≤5}Range :{1 ≤ x ≤ 6}
Check It Out! Example 1 Continued
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When the relation is also a function, you can write the inverse of the function f(x) as f–1(x). This notation does not indicate a reciprocal.
Functions that undo each other are inverse functions.
To find the inverse function, use the inverse operation. In the example above, 6 is added to x in f(x), so 6 is subtracted to find f–1(x).
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Use inverse operations to write the inverse of f(x) = x – if possible.
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Example 2: Writing Inverses of by Using Inverse Functions
is subtracted from the variable, x.12f(x) = x – 1
2
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Example 2 Continued
Substitute 1 for x.
Check Use the input x = 1 in f(x).
f(x) = x – 12
12Substitute for x.
Substitute the result into f–1(x)12f–1(x) = x +
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Use inverse operations to write the inverse of f(x) = .
x3 f(x) =
Check It Out! Example 2a
The variable x, is divided by 3.
x3
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Substitute 1 for x.
Check Use the input x = 1 in f(x).
13Substitute for x.
Substitute the result into f–1(x)
Check It Out! Example 2a Continued
f(x) = x3
f–1(x) = 3x
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Undo operations in the opposite order of the order of operations.
The reverse order of operations: Addition or Subtraction Multiplication or Division Exponents Parentheses
Helpful Hint
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Homework: 4-2 Practice A Worksheet