Chapter 1 | Functions and Graphs
21
The composite function g ∘ f is defined such that ⎛ ⎝ g ∘ f ⎞ ⎠ ( x ) = g ⎛ ⎝ f ( x ) ⎞
⎠ =3 f ( x )+1=3 x 2 +1.
Note that these two new functions are different from each other. Combining Functions with Mathematical Operators To combine functions using mathematical operators, we simply write the functions with the operator and simplify. Given two functions f and g , we can define four new functions: ⎛ ⎝ f + g ⎞ ⎠ ( x ) = f ( x )+ g ( x ) Sum ⎛ ⎝ f − g ⎞ ⎠ ( x ) = f ( x )− g ( x ) Difference ⎛ ⎝ f · g ⎞ ⎠ ( x ) = f ( x ) g ( x ) Product ⎛ ⎝ f g ⎞ ⎠ ( x ) = f ( x ) g ( x ) for g ( x ) ≠0 Quotient
Example 1.6 Combining Functions Using Mathematical Operations
Given the functions f ( x ) =2 x −3 and g ( x ) = x 2 −1, find each of the following functions and state its domain.
a. ( f + g )( x ) b. ( f − g )( x ) c. ( f · g )( x ) d. ⎛ ⎝ f g ⎞ ⎠ ( x )
Solution a. ⎛
⎝ f + g ⎞ ⎠ ( x ) = (2 x −3)+( x 2 −1) = x 2 +2 x −4. The domain of this function is the interval (−∞, ∞). b. ⎛ ⎝ f − g ⎞ ⎠ ( x ) = (2 x −3)−( x 2 −1) =− x 2 +2 x −2. The domain of this function is the interval (−∞, ∞). c. ⎛ ⎝ f · g ⎞ ⎠ ( x ) = (2 x −3)( x 2 −1) =2 x 3 −3 x 2 −2 x +3. The domain of this function is the interval (−∞, ∞). d. ⎛ ⎝ f g ⎞ ⎠ ( x ) = 2 x −3 x 2 −1 . The domain of this function is { x | x ≠±1}.
For f ( x ) = x 2 +3 and g ( x ) =2 x −5, find ⎛ ⎝ f / g ⎞
1.4
⎠ ( x ) and state its domain.
Function Composition When we compose functions, we take a function of a function. For example, suppose the temperature T on a given day is described as a function of time t (measured in hours after midnight) as in Table 1.1 . Suppose the cost C , to heat or cool a building for 1 hour, can be described as a function of the temperature T . Combining these two functions, we can describe
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