The function concept is one of the central concepts in all of mathematics (Knuth, 2000; Romberg, Carpernter, & Fennema, 1993; Yerushalmy & Schwartz, 1993).
Understanding multiple representations of functions and the ability to move between them is critical to mathematical development (Knuth, 2000; Rider, 2007).
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Piecewise Functions A piecewise function is a function that is a combination of one or more functions.
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Read this as “ f of x is 5 if x is greater than 0 and less than 13 , 9 if x is greater than or equal to 13 and less than 55 , and 6.5 if x is greater than or equal to 55 . ” The rule for a piecewise function is different for different parts, (or pieces), of the domain (x-values) For instance, movie ticket prices are often different for different age groups. So the function for movie ticket prices would assign a different value (ticket price) for each domain interval (age group).
Looking at only “part” or a “piece” of the function What rule would you write for this function? (How could we restrict the original function?) f(x) = x 2 - 3 if x ≥ -2 x ≥ -2 f(x) = x 2 - 3
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Restricting the domain of a function What is the domain? All real numbers What is the equation for this graph? f(x) = –2x – 5
Looking at only “part” or a “piece” of the function What rule would you write for this function? f(x) = –2x – 5 f(x) = –2x – 5 if x < –2
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What rule would you write for this piecewise function? Piecewise Functions x 2 – 3 if x ≥ –2
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a) What is the value of y when x = –4? Give two ways to find it. Piecewise Functions b) Which equation would you use to find the value of y when x = 2? c) Which equation would you use to find the value of y when x = –2? x 2 – 3 if x ≥ –2
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Piecing it all together: Evaluating Piecewise Functions
Find the interval of the domain that contains the x-value
Then use the rule for that interval.
9 3 -1 25
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2x + 1 if x ≤ 2 x 2 – 4 if x > 2 h(x) = Because –1 ≤ 2, use the rule for x ≤ 2 . Because 4 > 2, use the rule for x > 2. h(–1) = 2(–1) + 1 = –1 h(4) = 4 2 – 4 = 12 Evaluate the piecewise function for: x = –1 and x = 4.
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3x 2 + 1 if x < 0 5x – 2 if x ≥ 0 g(x) = Because –1 < 0, use the rule for x < 0. Because 3 ≥ 0, use the rule for x ≥ 0. g(3) = 5(3) – 2 = 13 g(–1) = 3(–1) 2 + 1 = 4 Evaluate each piecewise function for: x = –1 and x = 3
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12 if x < –3 20 if x ≥ 6 f(x) = Because –3 ≤ –1 < 6 , use the rule for – 3 ≤ x < 6 f(–1) = 15 Evaluate each piecewise function for: x = –1 and x = 3 15 if –3 ≤ x < 6 f(3) = 15 Because –3 ≤ 3 < 6 , use the rule for – 3 ≤ x < 6
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