270 Chapter 3 Linear Equations in Two Variables Student Check 3 Use the vertical line test to determine if each relation is a function. a. b. 4 2 y –4 –2 2 4 –4 x c. d. ������������ All lines, except vertical lines, pass the vertical line test. Therefore, every linear equation of the form y = mx + b is a function. Function Notation An example of a linear equation that is a function is y = 4x - 5. This equation states a rule for writing ordered pairs where each x-coordinate is paired with exactly one y-coordinate. Function notation is a convenient way to determine this pairing of coordinates. Suppose we want to find the y-value that corresponds to x = 3 for y = 4x - 5. We determine the y-value by replacing the variable x with 3. y = 4x - 5 y = 4(3) -5 y = 12 - 5 = 7 So, the output value of 4x - 5 is 7 when x = 3. In other words, the point (3, 7) lies on the graph of y = 4x - 5. In mathematics, we use letters such as f, g, and h to name functions. The symbol f (x) is read “f of x” and means “the function of x.” This notation is called function notation. ������������������������ Function Notation The notation f (x) is used to denote that y is a function of x. y = f (x) Name of function Output value Input value In this notation, x is the input value and f (x) is the output value. We can use function notation to write an equivalent equation to y = 4x - 5, which is f (x) = 4x - 5. The equations are equivalent because y = f (x). The notation f (3) means to replace x with 3 and find the corresponding y-value. For example, f (x) = 4x - 5 f (3) = 4(3) - 5 = 7 4 1 8 2 –4 –2 –1 –8 x y 2 2 4 4 –2 –4 –2 –4 x y 2 2 4 4 –2 –4 –2 –4 x y Objective 4 ▶ Use function notation.
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