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4.1 Inverses Fri Oct 18

4.1 Inverses Fri Oct 18. Do Now Solve for Y 1) 2). Quiz Review. Retakes?. Inverses. When we go from an output of a function back to its inputs, we get an inverse relation Interchanging the first and second coordinates of each ordered pair produces the inverse relation. Notation.

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4.1 Inverses Fri Oct 18

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  1. 4.1 InversesFri Oct 18 Do Now Solve for Y 1) 2)

  2. Quiz Review • Retakes?

  3. Inverses • When we go from an output of a function back to its inputs, we get an inverse relation • Interchanging the first and second coordinates of each ordered pair produces the inverse relation

  4. Notation • The notation for the inverse of f(x) is • This is not an exponent!

  5. One-to-one Functions • A function f is one-to-one if different inputs have different outputs • If • Every y-value is unique

  6. Properties of one-to-one functions • If a function f is one-to-one, its inverse is a function • The domain of a one-tone function f is the range of its inverse • The range of a one-to-one function is the domain of its inverse • A function that is always increasing or decreasing is one-to-one

  7. Ways to show a function is one-to-one • 1) Assume f(a) = f(b); then show that a = b • If you can think of a y-value that has 2 x-values (ex: x^2 • 2) Horizontal Line Test • If a horizontal line intersects the graph more than once, it is NOT one-to-one

  8. How to find a formula for inverse • 1) Replace f(x) with y (if possible) • 2) Switch x and y • 3) Solve for y • 4) Replace y with

  9. Ex • Find an equation for the inverse of the relation

  10. Ex2 • Find an inverse for the function

  11. You try • Find an inverse for the following • 1) f(x) = 7 - x • 2) • 3)

  12. Closure • Find the inverse for the function • HW: p.356 #17-59 odds

  13. 4.1 Inverses and CompositionsMon Oct 21 • Do Now • Find the inverse of

  14. HW Review: p.356 #17-59 odds

  15. Inverses and Graphs • The graph of an inverse function is a reflection of f(x)’s graph across the line y = x

  16. Inverse Functions and Compositions • If a function f(x) is one-to-one, then the following compositions are true: for each x in the domain of f for each x in the domain of f inverse

  17. Ex • Given f(x) = 5x + 8, find its inverse and show they are inverses using composition

  18. Ex • Given , find its inverse and show they are inverses using composition

  19. Restricting the domain • In the case in which the inverse of a function is not a function, the domain can be restricted to allow the inverse to be a function • This is why square root graphs only show half of the true graph in a calculator

  20. Closure • Find the inverse of and show they are inverses using compositions HW: p.358 #67-87

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