Chapter 1 | Functions and Graphs
13
which implies
⎠ 2 − 2
x = 1 3 ⎛
⎝ y +1 ⎞
3 . We just need to verify that x is in the domain of f . Since the domain of f consists of all real numbers greater than or equal to −2/3, and
1 3 2 3 , there does exist an x in the domain of f . We conclude that the range of f is ⎧ ⎩ ⎛ ⎝ y +1 ⎞ ⎠ 2 − 2 3 ≥ −
⎨ y | y ≥−1 ⎫ ⎭ ⎬ .
c. Consider f ( x ) =3/( x −2). i. Since 3/( x −2) is defined when the denominator is nonzero, the domain is { x | x ≠2}. ii. To find the range of f , we need to find the values of y such that there exists a real number x in the domain with the property that 3 x −2 = y . Solving this equation for x , we find that x = 3 y +2. Therefore, as long as y ≠0, there exists a real number x in the domain such that f ( x ) = y . Thus, the range is ⎧ ⎩ ⎨ y | y ≠0 ⎫ ⎭ ⎬ .
1.2
Find the domain and range for f ( x ) = 4−2 x +5.
Representing Functions Typically, a function is represented using one or more of the following tools:
• A table • A graph • A formula We can identify a function in each form, but we can also use them together. For instance, we can plot on a graph the values from a table or create a table from a formula. Tables Functions described using a table of values arise frequently in real-world applications. Consider the following simple example. We can describe temperature on a given day as a function of time of day. Suppose we record the temperature every hour for a 24-hour period starting at midnight. We let our input variable x be the time after midnight, measured in hours, and the output variable y be the temperature x hours after midnight, measured in degrees Fahrenheit. We record our data in Table 1.1 .
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