1 / 20

Precalculus 1.7

Precalculus 1.7. INVERSE FUNCTIONS. DO THIS NOW!. You have a function described by the equation: f(x ) = x + 4 The domain of the function is: {0, 2, 5, 10} YOUR TASK: write the set of ordered pairs that would represent this function

bluma
Download Presentation

Precalculus 1.7

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Precalculus 1.7 INVERSE FUNCTIONS

  2. DO THIS NOW! • You have a function described by the equation: f(x) = x + 4 • The domain of the function is: {0, 2, 5, 10} • YOUR TASK: write the set of ordered pairs that would represent this function • **use the equation to find f(x) for each number in the domain • Graph the points in the function.

  3. Functions as Sets of Ordered Pairs • Recall that besides describing functions with equations or graphs, we can do so by listing the ordered pairs that make up the function

  4. Inverse function • Notation: f-1 is the inverse of f • Definition: A function’s inverse brings output values back to their input values. f(x)=x+4 6 2 INPUT OUTPUT f-1(x)=x-4

  5. Inverse Functions: 3 representations GRAPH ALGEBRA ORDERED PAIRS

  6. Inverse Functions: Ordered Pairs • Original function, f: {(0, 4) (2, 6) (5, 9) (10, 14)} • Inverse function, f-1: {(4, 0) (6, 2) (9, 5) (14, 10)} • What is f(2)? • What is f-1(6)? • To find the inverse function, represented by ordered pairs, simply flip each ordered pair • If f contains (x, y), then f-1 contains (y, x).

  7. Inverse Functions: Algebra • The equation of the inverse function should “undo” the equation of the original function. • Ex: If f(x) = x + 4, then f-1(x) = x – 4 • Ex: If g(x) = 4x, then g-1(x) = …?

  8. Precise Definition 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. Then g is the inverse of the function f. g = f-1 The domain of f is the range of f-1. The range of f is the domain of f-1.

  9. A logical point: If f is the inverse of g, then g is the inverse of f. Furthermore, g and f are inverses of each other.

  10. Example • Which of these is the inverse of:

  11. Inverse Function: GRAPHS • Graph these functions and their inverses on the same graph. • f(x) = x + 4 and g(x) = x – 4 • f(x) = 4x and g(x) = x/4 • f(x) = .5(x+3) and g(x) = 2x – 3 • f(x) = x2 and • {(0,1) (2,5) (3,6) (4,8)} and {(1,0) (5,2) (6,3) (8,4)}

  12. Inverse Graphs • The graph of a f and f-1 are related in a special way. • If (x,y) is on f’s graph, (y,x) must be on f-1’s graph. • Therefore, the graph of f-1 is a reflection of the graph of f across the line y = x.

  13. Practice • What is the inverse of: f(x) = 2x + 4 • What is the inverse of: f(x) = 1/x

  14. Which functions have inverses? What is this function’s inverse? (0, 1) (2, 4) (3, 4)

  15. Functions without inverses • If multiple input values have the same output value, then the function has no inverse • This is because given a repeated output, there would be no way to tell what the original input was • f: (0,1) (2,4) (3,4) • f-1: (1,0) (4,2) (4,3) This is NOT a function! Therefore we have no surefire way to undo f.

  16. Functions without inverses: a list • f(x) = x2 (What was x if f(x)=4?) • f(x) = xn where n is even • f(x) = |x|

  17. Functions without inverses: graph test If a function, f, has an inverse, f-1, then the inverse is also a function. Therefore f-1 must pass the vertical line test. In order for f-1 to pass the vertical line test, f must pass the HORIZONTAL LINE TEST. f-1 f

  18. Will these functions have inverses? 1) 2) 3) 4) If a function both increases and decreases, can it have an inverse?

  19. Finding the inverse of a function algebraically 1. 2. 3. 4. 5. • Use the horizontal line test to decide whether f has an inverse. • In the equation, replace f(x) with “y.” • Switch “x” and “y.” • Solve for y. • Replace y with “f-1(x).” • Check your work!

  20. Practice: find the inverses of these functions 2) 1) 3) 4) 5) 6)

More Related