Chapter 1 | Functions and Graphs
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Here, the notation −∞ refers to negative infinity, and it indicates that we are including all numbers less than or equal to zero, no matter how small. The set (−∞, ∞) = ⎧ ⎩ ⎨ x | x is any real number ⎫ ⎭ ⎬ refers to the set of all real numbers. Some functions are defined using different equations for different parts of their domain. These types of functions are known as piecewise-defined functions . For example, suppose we want to define a function f with a domain that is the set of all real numbers such that f ( x ) =3 x +1 for x ≥2 and f ( x ) = x 2 for x <2. We denote this function by writing
⎧ ⎩ ⎨ 3 x +1 x ≥2 x 2 x <2 .
f ( x ) =
When evaluating this function for an input x , the equation to use depends on whether x ≥2 or x <2. For example, since 5>2, we use the fact that f ( x ) =3 x +1 for x ≥2 and see that f (5) = 3(5) + 1 = 16. On the other hand, for x =−1, we use the fact that f ( x ) = x 2 for x <2 and see that f (−1) =1.
Example 1.1 Evaluating Functions
For the function f ( x ) =3 x 2 +2 x −1, evaluate a. f (−2) b. f ( 2) c. f ( a + h )
Solution Substitute the given value for x in the formula for f ( x ). a. f (−2) = 3(−2) 2 +2(−2)−1=12−4−1=7 b. f ( 2) =3( 2) 2 +2 2−1=6+2 2−1=5+2 2
f ( a + h ) =3( a + h ) 2 +2( a + h )−1 =3 ⎛
⎝ a 2 +2 ah + h 2 ⎞ ⎠ +2 a +2 h −1 =3 a 2 +6 ah +3 h 2 +2 a +2 h −1
c.
For f ( x ) = x 2 −3 x +5, evaluate f (1) and f ( a + h ).
1.1
Example 1.2 Finding Domain and Range
For each of the following functions, determine the i. domain and ii. range.
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