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Understanding Inverses of Relations and Functions

Learn how to graph, identify, and find inverses of relations and functions by reflecting points, switching x and y values, and using inverse operations. Practice writing inverse functions and understanding the multiplicative inverse.

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Understanding 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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