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Adding and subtracting functions The function f - g and its graph . Suppose that f and g are two functions defined on the interval [ ] , a b such that ( ) f x > ( ) g x for a x b . Then we can define a function h on the interval [ ] , a b by = ( ) h x - ( ) f x ( ) g x for all numbers x with a x b . At a given number c we can visualize the value of h as the distance between the point ( ) , c ( ) g c on the graph of g and the point ( ) , c ( ) f c vertically above it as indicated in the following picture. The function h is denoted by f - g so that (f - g) = ( ) x - ( ) f x ( ) g x for all numbers x such that a x b . Because of the condtion that ( ) f x > ( ) g x for a x b , the graph of the function f - g is above the x axis for a x b . Given any two real valued functions f and g, we can define a function f - g by the formula: (f - g) = ( ) x - ( ) f x ( ) g x . In order for - ( ) f x ( ) g x to be defined for a real number x it is necessary that both ( ) f x and ( ) g x are defined. It follows that the domain of the function f - g must consist of real numbers that are common to the domains of both of the functions f and g, that is, it is the intersection of the domains of f and g. We have D(f - g) = D(f) D(g). Example 1 : Let f and g be the two functions defined by = ( ) f x - 4 x 2 and = ( ) g x 3 x respectively. (a) State the domains of f, g and f - g. (b) Sketch the graph of f, g and f - g.

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Solution : (a) The domain of f is the interval { x | - 2 x 2 } = [ ] , - 2 2 , while the domain of g is the interval { x | x > 0 } = [ , 0 ). The intersection of these two domains is the interval { x | 0 x 2 } = [ ] , 0 2 , which is the domain of f - g. (b) The graph of f is the part of the circle with its centre at the origin and radius 2 which lies on or above the x axis. Since = 3 x 3 x for all real numbers x with x > 0, the graph of g is obtained by stretching the graph of = y x parallel to the y axis with a scale factor of 3. In the following picture, the graph of = y - 4 x 2 , that is, the graph of f, is drawn in purple , while the graph of = y 3 x , that is, the graph of g, is drawn in green . It looks as though the two graphs intersect where
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