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Understanding Inverse Functions and Their Properties

Learn about inverse functions in math, including relations, equations, tables, and graphs. Discover how to find and verify inverse functions step by step. Explore the importance of passing vertical and horizontal line tests.

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Understanding Inverse Functions and Their Properties

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  1. Inverse Functions

  2. Review from Math I • Relation – a mapping of input values (x-values) onto output values (y-values). • Here are 3 ways to show the same relation. x y -2 4 -1 1 0 0 1 1 y = x2 Equation Table of values Graph

  3. x y • -2 • -1 • 0 0 • 1 1 x = y2 • Inverse relation – just think: switch the x & y-values. ** the inverse of an equation: switch the x & y and solve for y. ** the inverse of a table: switch the x & y. ** the inverse of a graph: the reflection of the original graph in the line y = x.

  4. Ex: Find an inverse of y = -3x+6. • Steps: -switch x & y -solve for y y = -3x+6 x = -3y+6 x-6 = -3y

  5. Inverse Functions • Given 2 functions, f(x) & g(x), if f(g(x))=x AND g(f(x))=x, then f(x) & g(x) are inverses of each other. Symbols: f -1(x) means “f inverse of x”

  6. Ex: Verify that f(x)=-3x+6 and g(x)=-1/3x+2 are inverses. • Meaning find f(g(x)) and g(f(x)). If they both equal x, then they are inverses. f(g(x))= -3(-1/3x+2)+6 = x-6+6 = x g(f(x))= -1/3(-3x+6)+2 = x-2+2 = x ** Because f(g(x))=x and g(f(x))=x, they are inverses.

  7. To find the inverse of a function: • Change the f(x) to a y. • Switch the x & y values. • Solve the new equation for y. ** Remember functions have to pass the vertical line test!

  8. Ex: (a)Find the inverse of f(x)=x5. (b) Is f -1(x) a function? (hint: look at the graph! Does it pass the vertical line test?) • y = x5 • x = y5 Yes , f -1(x) is a function.

  9. Horizontal Line Test • Used to determine whether a function’s inverse will be a function by seeing if the original function passes the horizontal line test. • If the original function passes the horizontal line test, then its inverse is a function. • If the original function does not pass the horizontal line test, then its inverse is not a function.

  10. Ex: Graph the function f(x)=x2 and determine whether its inverse is a function. Graph does not pass the horizontal line test, therefore the inverse is not a function.

  11. Ex: f(x)=2x2-4 Determine whether f -1(x) is a function, then find the inverse equation. y = 2x2-4 x = 2y2-4 x+4 = 2y2 OR, if you fix the tent in the basement… f -1(x) is not a function.

  12. Ex: g(x)=2x3 y=2x3 x=2y3 OR, if you fix the tent in the basement… Inverse is a function!

  13. Assignment#1-16 EVENS ONLY, show work for all & check answers at the of the link:http://cdn.kutasoftware.com/Worksheets/Alg2/Function%20Inverses.pdf

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