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Chapter 1 | Functions and Graphs
the cost of heating or cooling a building as a function of time by evaluating C ⎛ ⎝ T ( t ) ⎞ ⎠ . We have defined a new function, denoted C ∘ T , which is defined such that ( C ∘ T )( t ) = C ( T ( t )) for all t in the domain of T . This new function is called a composite function. We note that since cost is a function of temperature and temperature is a function of time, it makes sense to define this new function ( C ∘ T )( t ). It does not make sense to consider ( T ∘ C )( t ), because temperature is not a function of cost. Definition Consider the function f with domain A and range B , and the function g with domain D and range E . If B is a subset of D , then the composite function ( g ∘ f )( x ) is the function with domain A such that (1.1) ⎛ ⎝ g ∘ f ⎞ ⎠ ( x ) = g ⎛ ⎝ f ( x ) ⎞ ⎠ . A composite function g ∘ f can be viewed in two steps. First, the function f maps each input x in the domain of f to its output f ( x ) in the range of f . Second, since the range of f is a subset of the domain of g , the output f ( x ) is an element in the domain of g , and therefore it is mapped to an output g ⎛ ⎝ f ( x ) ⎞ ⎠ in the range of g . In Figure 1.12 , we see a visual image of a composite function.
Figure 1.12 For the composite function g ∘ f , we have ⎛ ⎝ g ∘ f ⎞ ⎠ (1) =4, ⎛ ⎝ g ∘ f ⎞ ⎠ (2) =5, and ⎛ ⎝ g ∘ f ⎞ ⎠ (3) =4.
Example 1.7 Compositions of Functions Defined by Formulas
Consider the functions f ( x ) = x 2 +1 and g ( x ) =1/ x . a. Find ( g ∘ f )( x ) and state its domain and range. b. Evaluate ( g ∘ f )(4), ( g ∘ f )(−1/2). c. Find ( f ∘ g )( x ) and state its domain and range. d. Evaluate ( f ∘ g )(4), ( f ∘ g )(−1/2).
Solution a. We can find the formula for ( g ∘ f )( x ) in two different ways. We could write ( g ∘ f )( x ) = g ( f ( x )) = g ( x 2 +1) = 1 x 2 +1 .
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