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Section 2.6 Inverse Functions

Section 2.6 Inverse Functions. What you should learn. How to find inverse functions informally and verify that two functions are inverse functions of each other How to use graphs of functions to determine whether functions have inverse functions

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Section 2.6 Inverse Functions

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  1. Section 2.6 Inverse Functions

  2. What you should learn • How to find inverse functions informally and verify that two functions are inverse functions of each other • How to use graphs of functions to determine whether functions have inverse functions • How to use the horizontal line test to determine if functions are one-to-one • How to find inverse functions algebraically

  3. Consider the function, f(x) that doubles x and then subtracts 4. Now swap x and y to create a new function g(x) This function needs to add 4 first, then divide by 2

  4. Inverse Notation • f -1(x) is read as the f-inverse • Notice that an inverse of a function does the opposite thing in the opposite order of the original function. So g(x) is the inverse of f(x). We don’t use g(x). We use the inverse notation.

  5. Composition of Inverses Composition is something we did in section 2.5. What is the name of this function? This is the identity function.

  6. Definition of Inverse Function Let f and g be two functions such that f(g(x))=x for every x in the domain of g and g(f(x))=x for every x in the domain of f. Under these conditions the function g is the inverse function of the function f.

  7. Verifying Inverse Functions • To verify that two functions f and g are inverse functions, form the composition. • If the composition yields the identity function then the two functions are inverses of each other. Therefore g(x) is not the inverse of f(x).

  8. Graphs of Inverse Functions

  9. Vertical Line Test?

  10. Vertical Line Test? The inverse relation does not pass the vertical line test. The inverse is not a function.

  11. Horizontal Line Test • A function f has an inverse function if and only if no horizontal line intersects the graph of f at more than one point. The inverse relation will not be a function.

  12. One-to-One • A function f is one-to-one if each value of the dependent variable corresponds to exactly one value of the independent variable. • A function f has an inverse function if and only if f is one-to-one. • In other words it must pass both a vertical and horizontal line tests

  13. One-to-One? The inverse will be a function.

  14. One-to-One? The inverse will not be a function.

  15. One-to-One? The inverse will be a function.

  16. One-to-One? The inverse will not be a function.

  17. One-to-One? FAIL Vertical Line Test PASS Horizontal Line Test NOT One-to-One

  18. One-to-One? Pass Vertical Line Test PASS Horizontal Line Test One-to-One

  19. Finding Inverse Functions Algebraically • Use the horizontal line test to decide whether f has an inverse function. • Replace f(x) with y. • Swap x and y, and solve for y. • Replace y with f -1(x). • Verify that the range of one is the domain of the other.

  20. #58 page 240 Produce an inverse. • Replace f(x) with y. • Swap x and y, and solve for y. • Replace y with f -1(x).

  21. Find f-1(x) Replace f(x) with y Swap x with y Solve for y Replace y with f -1 (x)

  22. Homework • Page 239 • 1-4, 13-17, 39-44

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