SECTION 2.4 Library of Functions; Piecewise-defined Functions 105 The functions discussed so far are basic. Whenever you encounter one of them, you should see a mental picture of its graph. For example, if you encounter the function f x x ,2 ( ) = you should see in your mind’s eye a picture of a parabola. Now Work PROBLEMS 11 THROUGH 18 2 Analyze a Piecewise-defined Function Sometimes a function is defined using different equations on different parts of its domain. For example, the absolute value function f x x ( ) = is actually defined by two equations: f x x ( ) = if x 0 ≥ and f x x ( ) = − if x 0. < See Figure 50. For convenience, these equations are generally combined into one expression as ( ) = = ≥ − < ⎧ ⎨ ⎪⎪ ⎩⎪⎪ f x x x x x x if 0 if 0 When a function is defined by different equations on different parts of its domain, it is called a piecewise-defined function. Analyzing a Piecewise-defined Function A piecewise-defined function f is defined as ( ) = − < ≥ ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ f x x x x x 2 if 0 if 0 (a) Find f 4 , ( ) − f 0 , ( ) and f 4 . ( ) (b) Find the domain of f. (c) Locate any intercepts. (d) Graph f. (e) Use the graph to find the range of f. Solution EXAMPLE 3 (a) When evaluating a piecewise-defined function, we first must identify which equation should be used in the evaluation. We do this by identifying in which part of the domain the value for the independent variable lies. • To find f 4 , ( ) − observe that 4 0 − < so x 4 = − lies in the domain of the first equation. This means that when x 4, = − the equation for f is f x x2 . ( ) = − Then f 4 2 4 8 ( ) ( ) − = − − = • To find f 0 , ( ) observe that when x 0, = the equation for f is f x x. ( ) = Then f 0 0 0 ( ) = = • To find f 4 , ( ) observe that when x 4, = the equation for f is f x x. ( ) = Then f 4 4 2 ( ) = = (b) The domain of the first equation of the function f is the set of all negative real numbers, { } < x x 0 , or , 0 ( ) −∞ in interval notation.The domain of the second equation of the function f is the set of all nonnegative real numbers, x x 0 , { } ≥ or 0, [ )∞ in interval notation. The domain of the function is the union of the domains of the individual equations. Since , 0 0, , ( ) [ ) ( ) −∞ ∪ ∞ = −∞ ∞ the domain of f is the set of all real numbers. Figure 50 Absolute Value Function x y 3 3 23 (1, 1) (0, 0) (21, 1) (2, 2) (22, 2) f(x) = ƒ x ƒ (continued)
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