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6.7 Inverse functions

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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6.7 Inverse functions

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  1. 6.7 Inverse functions

  2. 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

  3. 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)

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

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