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Inverses of Relations and Functions

Inverses of Relations and Functions. Essential Questions. How do we graph and recognize inverses of relations and functions? How do we find inverses of functions?. Holt McDougal Algebra 2. Holt Algebra 2. The multiplicative inverse of 5 is. The multiplicative inverse matrix of.

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Inverses of Relations and Functions

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  1. Inverses of Relations and Functions Essential Questions • How do we graph and recognize inverses of relations and functions? • How do we find inverses of functions? Holt McDougal Algebra 2 Holt Algebra 2

  2. The multiplicative inverse of 5 is The multiplicative inverse matrix of You have seen the word inverse used in various ways. The additive inverse of 3 is –3.

  3. Remember! 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. You can 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.

  4. Example 1: Graphing Inverse Relations Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. Range: [2, 9] Domain: [0, 8] Graph each ordered pair and connect them. ● • Switch the x- and y-values in each ordered pair. ● • ● ● • • Range: [0, 8] Domain: [2, 9]

  5. Example 2: Graphing Inverse Relations Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. Range: [0, 5] Domain: [1, 6] Graph each ordered pair and connect them. Switch the x- and y-values in each ordered pair. • • • • • • • • • • Range: [1, 6] Domain: [0, 5]

  6. 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).

  7. Use inverse operations to write the inverse of f (x) = x – if possible. 1 is subtracted from the variable, x. 2 1 1 1 = 1 1 2 2 2 1 1 1 1 1 1 f(x) = x – f(1) = 1 – f(x) = x – Add to x to write the inverse. 2 2 2 2 2 2 2 2 f–1(x) = x + f–1(x) = x + f–1( ) = + The inverse function does undo the original function.  Example 3: Writing Inverses of by Using Inverse Functions Check Use the input x = 1 in f(x). Substitute the result into f–1(x) = 1

  8. Use inverse operations to write the inverse of f(x) =. x x 1 3 3 3 f(x) = 1 1 1 = x 3 3 3 3 f(x) = f(1) = f–1( ) = 3( ) The inverse function does undo the original function.  Example 4: Writing Inverses of by Using Inverse Functions The variable x, is divided by 3. f–1(x) = 3x Multiply by 3 to write the inverse. CheckUse the input x = 1 in f(x). Substitute the result into f–1(x) f–1(x) = 3x = 1

  9. Lesson 8.2 Practice A

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