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7.4 Inverse Functions

7.4 Inverse Functions. p. 422. Review from chapter 2. 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 = x 2. Equation Table of values Graph. x y -2

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7.4 Inverse Functions

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  1. 7.4 Inverse Functions p. 422

  2. Review from chapter 2 • 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!

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