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Section 1'9 Inverse Functions

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How to use the horizontal line test to determine if functions are one-to-one. How to find inverse functions algebraically ... Finding Inverse Functions Algebraically ... – PowerPoint PPT presentation

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Title: Section 1'9 Inverse Functions


1
Section 1.9 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.

5
Composition of Inverses
What is the name of this 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.
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 in inverse function if and only
    if f is one-to-one.

13
One-to-One?
14
One-to-One?
15
One-to-One?
16
One-to-One?
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
Find f -1(x)
Replace f(x) with y
Swap x with y
Solve for y
Replace y with f -1 (x)
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