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Inverse Functions and their Representations
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Definition A function is a set of ordered pairs with no two first elements alike. f(x) = { (x,y) : (3, 2), (1, 4), (7, 6), (9,12) } But ... what if we reverse the order of the pairs? This is also a function ... it is the inverse function f -1(x) = { (x,y) : (2, 3), (4, 1), (6, 7), (12, 9) }
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Example Consider an element of an electrical circuit which increases its resistance as a function of temperature. T = Temp R = Resistance -20 50 150 20 250 40 350 R = f(T)
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Now we would say that g(R) and f(T) are inverse functions
Example We could also take the view that we wish to determine T, temperature as a function of R, resistance. R = Resistance T = Temp 50 -20 150 250 20 350 40 T = g(R) Now we would say that g(R) and f(T) are inverse functions
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For example, let’s take a look at the square function: f(x) = x2
In the same way, the inverse of a given function will “undo” what the original function did. For example, let’s take a look at the square function: f(x) = x2 x f(x) y f--1(x) 9 3 3 9 9 3 3 9 9 3 3 9 9 3 3 x2 9 9 3 3 9 9 9 3 3 3 9 9
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For example, let’s take a look at the square function: f(x) = x2
In the same way, the inverse of a given function will “undo” what the original function did. For example, let’s take a look at the square function: f(x) = x2 x y f--1(x) f(x) 5 25 5 5 5 25 25 5 5 25 25 5 5 x2 25 5 5 25 5 25 25 5 25 5 5 5
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For example, let’s take a look at the square function: f(x) = x2
In the same way, the inverse of a given function will “undo” what the original function did. For example, let’s take a look at the square function: f(x) = x2 x f(x) y f--1(x) 11 121 11 11 11 121 121 11 11 121 121 11 11 x2 121 121 121 11 11 121 121 11 11 121 121 121 11 121 11
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Graphically, the x and y values of a point are switched.
The point (4, 7) has an inverse point of (7, 4) AND The point (-5, 3) has an inverse point of (3, -5)
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Graphically, the x and y values of a point are switched.
If the function y = g(x) contains the points x 1 2 3 4 y 8 16 then its inverse, y = g-1(x), contains the points x 1 2 4 8 16 y 3 Where is there a line of reflection?
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y = f(x) y = x The graph of a function and its inverse are mirror images about the line y = f-1(x) y = x
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Find the inverse of a function :
Example 1: y = 6x - 12 Step 1: Switch x and y: x = 6y - 12 Step 2: Solve for y:
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Example 2: Given the function : y = 3x2 + 2 find the inverse:
Step 1: Switch x and y: x = 3y2 + 2 Step 2: Solve for y:
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Definition of an Inverse Function
A function, f, has an inverse function, g, if and only if f(g(x)) = x and g(f(x)) = x, for every x in domain of g and in the domain of f.
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