239 2.4 Linear Functions If a = 0 in the definition of linear function, then the equation becomes ƒ1x2 = b. In this case, the domain is 1-∞, ∞2 and the range is 5b6. A function of the form ƒ1x2 =b is a constant function, and its graph is a horizontal line. EXAMPLE 3 Graphing a Vertical Line Graph x = -3. Give the domain and range of this relation. SOLUTION Because x always equals -3, the value of x can never be 0, and the graph has no y-intercept. Using reasoning similar to that of Example 2, we find that this graph is parallel to the y-axis, as shown in Figure 35. The domain of this relation, which is not a function, is 5-36, while the range is 1-∞, ∞2. S Now Try Exercise 25. x y (–3, 0) 0 x = –3 Vertical line Figure 35 Standard Form Ax +By =C Equations of lines are often written in the form Ax +By =C, known as standard form. EXAMPLE 2 Graphing a Horizontal Line Graph ƒ1x2 = -3. Give the domain and range. SOLUTION Because ƒ1x2, or y, always equals -3, the value of y can never be 0 and the graph has no x-intercept. If a straight line has no x-intercept then it must be parallel to the x-axis, as shown in Figure 33. The domain of this linear function is 1-∞, ∞2. The range is 5-36. Figure 34 shows the calculator graph. y x 0 1 f(x) = –3 (0, –3) Horizontal line Figure 33 y1 = −3 −10 −10 10 10 Figure 34 S Now Try Exercise 17. NOTE The definition of “standard form” is, ironically, not standard from one text to another. Any linear equation can be written in infinitely many different, but equivalent, forms. For example, the equation 2x + 3y = 8 can be written equivalently as 2x + 3y - 8 = 0, 3y = 8 - 2x, x + 3 2 y = 4, 4x + 6y = 16, and so on. In this text we will agree that if the coefficients and constant in a linear equation are rational numbers, then we will consider the standard form to be Ax + By = C, where A Ú 0, A, B, and C are integers, and the greatest common factor of A, B, and C is 1. If A = 0, then we choose B70. (If two or more integers have a greatest common factor of 1, they are said to be relatively prime.)
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