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6.7 Inverse functions. What is an inverse function?. Inverse Relation: If an relation pairs of element of a of its domain and b of its range pairs b with a. For example: if (a, b) is an ordered pair of a relation then (b, a) is an ordered pair of its inverse

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what is an inverse function
What is an inverse function?
  • Inverse Relation: If an relation pairs of element of a of its domain and b of its range pairs b with a. For example: if (a, b) is an ordered pair of a relation then (b, a) is an ordered pair of its inverse
  • Inverse function: if both a relation and its inverse are functions
find an inverse given a table or graph
Find an inverse given a table or graph
  • What is the inverse of the given relation?
  • Switch your x & y 1) plot the pts.

and rewrite 2) switch to (y,x)

3) plot (y, x)

finding the inverse of a function
Finding the inverse of a function
  • Change f(x) to y
  • Switch your x and y
  • Solve for y
  • Rewrite as f-1(x)
  • Determine if f-1(x) is a function
examples
examples

Find the inverse function and determine if it’s a function

(We will be using the index cards to the first 2 examples)

  • 1. f(x) = 3x + 5
  • 2. f(x) = 6x – 8
  • 3. f(x) = x2 + 4
  • 4. f(x) = 5 – 2x2
  • 5. f(x) = 4 – 3x
determine domain of inverse function
Determine domain of inverse function
  • Domain: all your x values

For any liner, quadratic, cubic, etc (where the exponent is a whole #) your domain is always:

all real numbers or (-∞, ∞)

If there is an even root (sq. root, 4th root etc.), you need to determine what value of x will make the expression = 0, that x value is the minimum domain value & will be written as [#, ∞)

Examples: use the 5 example we just did

determine the range
Determine the range
  • Range is all the y values
  • Determine if your function has a minimum or a maximum value (not both)
  • If there is not just 1 minimum or maximum your range is: all real numbers or (-∞,∞)
  • If there is just 1 minimum then your range is: [minimum y value, ∞)
  • If there is just 1 maximum then your range is:

(-∞, maximum y value]

- examples: let’s look at the 5 we did

graphing
Graphing
  • 2 ways: You must graph both the relation & its inverse
  • 1st way: by hand
    • A) make a table of values (w/ a minimum of 5 points – if linear pick 2 “-”, 0, and 2 “+”, if it’s quadratic find the vertex & then pick 2 < the vertex and 2 > the vertex)
    • B) plot each point and connect

2nd way: calculator

    • A) enter both graphs on the calculator
    • B) sketch what you see, make sure you have accurate points, so you may have to look at the table of values
composition of inverse functions
Composition of Inverse Functions
  • If f and f-1 are inverse:

(f-1˚ f)(x) = x and (f ˚ f-1)(x) = x for x in the domains of f and f-1 respectively

Examples: determine if the functions are inverses

  • f(x) = 10x – 10 and f-1(x) = x +10

10

2. f(x) = 3 – 7x and f-1(x) = x – 7

3