Chapter 1 | Functions and Graphs
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the formula changes with the sign of x :
⎧ ⎩ ⎨ − x , x <0 x , x ≥0 .
f ( x ) =
Other piecewise-defined functions may be represented by completely different formulas, depending on the part of the domain in which a point falls. To graph a piecewise-defined function, we graph each part of the function in its respective domain, on the same coordinate system. If the formula for a function is different for x < a and x > a , we need to pay special attention to what happens at x = a when we graph the function. Sometimes the graph needs to include an open or closed circle to indicate the value of the function at x = a . We examine this in the next example. Example 1.19 Graphing a Piecewise-Defined Function
Sketch a graph of the following piecewise-defined function:
⎧ ⎩ ⎨ x +3,
x <1
f ( x ) =
.
( x −2) 2 ,
x ≥1
Solution Graph the linear function y = x +3 on the interval (−∞, 1) and graph the quadratic function y = ( x −2) 2 on the interval [1, ∞). Since the value of the function at x =1 is given by the formula f ( x ) = ( x −2) 2 , we see that f (1) =1. To indicate this on the graph, we draw a closed circle at the point (1, 1). The value of the function is given by f ( x ) = x +2 for all x <1, but not at x =1. To indicate this on the graph, we draw an open circle at (1, 4).
Figure 1.22 This piecewise-defined function is linear for x <1 and quadratic for x ≥1.
1.14
Sketch a graph of the function
⎧ ⎩
⎨ 2− x , x ≤2 x +2, x >2 .
f ( x ) =
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