SECTION 2.1 Functions 65 It is usually easiest to determine whether an equation, where y depends on x, is a function when the equation is solved for y. If any value of x in the domain corresponds to more than one y, the equation does not define y as a function of x ; otherwise, it does define a function. Determining Whether an Equation Is a Function Determine whether the equation + = x y 1 2 2 defines y as a function of x. Solution EXAMPLE 5 To determine whether the equation + = x y 1, 2 2 which defines the unit circle, is a function, solve the equation for y. + = = − = ± − x y y x y x 1 1 1 2 2 2 2 2 For values of x for which − < < x 1 1, two values of y result. For example, if = x 0, then = ± y 1, so two different outputs result from the same input. This means that the equation + = x y 1 2 2 does not define y as a function of x. See Figure 12. Now Work PROBLEM 37 3 Use Function Notation; Find the Value of a Function It is common practice to denote functions by letters such as f g F G , , , , and others. If f is a function, then for each number x in the domain, the corresponding number y in the range is designated by the symbol ( ) f x , read as “ f of x,” and we write ( ) = y f x . When a function is expressed in this way, we are using function notation . We refer to ( ) f x as the value of the function f at the number x. For example, the function in Example 4 may be written using function notation as ( ) = = − y f x x2 5. Then ( ) = − f 1 3 and ( ) = f 3 1. Sometimes it is helpful to think of a function f as a machine that receives as input a number from the domain, manipulates it, and outputs a value in the range. See Figure 13. The restrictions on this input/output machine are as follows: • It accepts only numbers from the domain of the function. • For each input, there is exactly one output (which may be repeated for different inputs). Determining Whether an Equation Is a Function Determine whether the equation = − y x2 5 defines y as a function of x. Solution EXAMPLE 4 The equation tells us to take an input x, multiply it by 2, and then subtract 5. For any input x, these operations yield only one output y, so the equation is a function. For example, if = x 1, then = ⋅ − = − y 2 1 5 3. If = x 3, then = ⋅ − = y 2 3 5 1. Figure 11 = − y x2 5 x y 0 5 –5 (3, 1) (1, –3) 3 –5 The graph of the equation = − y x2 5 is a line with slope 2 and y -intercept −5. The function is called a linear function . See Figure 11. Figure 12 + = x y 1 2 2 x y 2 2 22 –2 0 1 –1 (0, 1) (0, –1) Figure 13 Input/output machine Input x Output y 5 f (x) f x In Words If ( ) = y f x , then x is the input and y is the output corresponding to x.
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