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Chapter 1 | Functions and Graphs
f ( x 1 ) < f ( x 2 )when x 1 < x 2 . We say that a function f is decreasing on the interval I if for all x 1 , x 2 ∈ I , f ( x 1 ) ≥ f ( x 2 ) if x 1 < x 2 . We say that a function f is strictly decreasing on the interval I if for all x 1 , x 2 ∈ I , f ( x 1 ) > f ( x 2 ) if x 1 < x 2 .
For example, the function f ( x ) =3 x is increasing on the interval (−∞, ∞) because 3 x 1 <3 x 2 whenever x 1 < x 2 . On the other hand, the function f ( x ) =− x 3 is decreasing on the interval (−∞, ∞) because − x 1 3 > − x 2 3 whenever x 1 < x 2 ( Figure 1.11 ).
Figure 1.11 (a) The function f ( x ) =3 x is increasing on the interval (−∞, ∞). (b) The function f ( x ) =− x 3 is decreasing on the interval (−∞, ∞).
Combining Functions Now that we have reviewed the basic characteristics of functions, we can see what happens to these properties when we combine functions in different ways, using basic mathematical operations to create new functions. For example, if the cost for a company to manufacture x items is described by the function C ( x ) and the revenue created by the sale of x items is described by the function R ( x ), then the profit on the manufacture and sale of x items is defined as P ( x ) = R ( x )− C ( x ). Using the difference between two functions, we created a new function. Alternatively, we can create a new function by composing two functions. For example, given the functions f ( x ) = x 2 and g ( x ) =3 x +1, the composite function f ∘ g is defined such that ⎛ ⎝ f ∘ g ⎞ ⎠ ( x ) = f ⎛ ⎝ g ( x ) ⎞ ⎠ = ⎛ ⎝ g ( x ) ⎞ ⎠ 2 = (3 x +1) 2 .
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