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| Operation on Real Functions |
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| Functions can be added, subtracted and multiplied. They can also be divided where the divisor function does not take the value zero. These operations create new functions. |
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| If f(x) and g(x) are two real valued functions, then for every value of x that belongs to both the domains of f and g, we can define the following functions: |
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| Sum Function: (f + g) (x) = f (x) + g (x) |
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| Difference Function: (f - g) (x) = f(x) - g(x) |
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| Product Function: fg(x) = f(x) g(x) |
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Domain of these functions are {Domain f}  {Domain g} |
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(c f) (x) = c.f (x) for all x  Domain (f) |
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| Example: |
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Let f(x) =  |
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g(x) =  |
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{Domain (f) = (0, ) |
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| f + g (x) = f(x) + g (x) Domain |
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 [0,1] |
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 [0,1] |
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 [0,1] |
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| If range(f) C Dom(g), we define the composite function of g and f (gof) by |
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gof(x) = g [f(x)] for all x A |
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| If range (g) C dom f, we define the composite function (fog) of f and g by |
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fog (x) = f [g(x)] for all x X |
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| Example: |
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| Let f(x) = x2-1, g(x) = 3x-1 |
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| Domain f = R |
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| Domain g = R |
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| fog (x) = f [g(x)] |
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| (range of g C domain of f ) |
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| = f(3x-1) |
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| = (3x-1)2 - 1 |
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| = 9x2 + 1 - 6x - 1 |
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| = 9x2 - 6x |
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| Domain fog = R |
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Let f : A B be a real valued one-one and onto function. |
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| Therefore, we can define a function, (denoted by f-1) called 'the inverse of f ' as follows: |
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f-1 : B A such that |
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| Example: |
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If f: R  R is defined by |
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| f(x) = 5x - 7, find f-1(x) and f-1(8). |
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| Suggested answer: |
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| Let f (x) = y, then y = 5x - 7 |
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| f is a one-one and onto function. |
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| This is inverse function of f. |
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