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mult_and_divide_functions

# mult_and_divide_functions - Multiplying and dividing...

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Multiplying and dividing functions The functions f . g and f/g . Let f and g be any two real valued functions. We can define a function f . g on a domain which consists of real numbers that are common to the domains of both of the functions f and g, that is, on the intersection of the domains of f and g, by the formula: (f . g) = ( ) x ( ) f x . ( ) g x . We have D(f . g) = D(f) D(g). We can also define a function f / g by the formula: (f / g) = ( ) x ( ) f x / ( ) g x . Since division by zero must be avoided, the domain of f / g consists of all real numbers x such that both ( ) f x and ( ) g x are defined and such that ( ) g x 0. Thus D(f / g) = { x | x D(f) and x D(g) and ( ) g x 0 } = { x | x D(f) D(g) and ( ) g x 0 }. Example 1 : Let f and g be the two functions defined by = ( ) f x + 4 x and = ( ) g x - 4 x respectively. (a) State the domains of f, g, f . g and f / g. (b) Give simplified formulas to describe the functions f . g and f / g, and sketch the graphs of these two functions. Solution : (a) The domain of f is the set of all real numbers that are greater than or equal to - 4, that is, D(f) = { x | x > -4 } = [ , - 4 ). The domain of g is the set of all real numbers that are less than or equal to 4, that is, D(g) = { x | x 4 } = ( , -∞ 4 ]. We have D(f . g) = { x | x > - 4 and x 4 } = { x | - 4 x 4 } = [ ] , - 4 4 . Since = ( ) g x 0 when = x 4, we have D( f / g ) = { x | - 4 x < 4 } = [ , - 4 4 ). (b) ( f . g ) = ( ) x + 4 x . - 4 x = ( ) + 4 x ( ) - 4 x = - 16 x 2 . The graph of f . g is the part of the circle with centre at the origin and radius 4 that lies on or above the x axis. ( f / g ) = ( ) x + 4 x - 4 x = + 4 x - 4 x .

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We can obtain an alternative expression for + 4 x - 4 x by performing the following polynomial division. -1 ______ - + x 4 | + x 4 - x 4 _____ 8 This division shows that = + 4 x - 4 x - + 1 8 - 4 x = - - 8 - x 4 1. The graph of = y - - 8 - x 4 1 is a rectangular hyperbola obtained by applying appropriate transformations to the graph of = y 1 x .
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mult_and_divide_functions - Multiplying and dividing...

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