If [tex]f(x)=\left\{\begin{array}{lll}-2 x-3 & \text { if } & x<3 \\ x^2-2 x & \text { if } & x \geq 3\end{array}\right.[/tex], what is [tex]f(t+2)[/tex] when [tex]t=2[/tex]?
A. -5
B. 0
C. -1
D. 8
Answer (2)
Calculate t + 2 = 2 + 2 = 4 .
Determine that x = 4 falls into the second case of the piecewise function, xg e q 3 .
Substitute x = 4 into the second case: f ( 4 ) = 4 2 − 2 ( 4 ) .
Calculate f ( 4 ) = 16 − 8 = 8 , so the final answer is 8 .
Explanation
Find t+2 First, we need to find the value of t + 2 when t = 2 .
Calculate t+2 Substituting t = 2 into t + 2 , we get 2 + 2 = 4 . So, we need to find f ( 4 ) .
Determine which case to use Now we need to determine which case of the piecewise function to use. Since 4 l ess 3 but 4 g e q 3 , we use the second case: f ( x ) = x 2 − 2 x .
Substitute x=4 Substitute x = 4 into f ( x ) = x 2 − 2 x to find f ( 4 ) .
Calculate f(4) f ( 4 ) = ( 4 ) 2 − 2 ( 4 ) = 16 − 8 = 8 .
Examples
Piecewise functions are used in real life to model situations where different rules or conditions apply over different intervals. For example, consider a cell phone plan where you pay a fixed monthly fee for a certain amount of data, and then you pay an additional fee for each gigabyte of data you use beyond that limit. This can be modeled as a piecewise function, where the cost is constant up to the data limit, and then increases linearly with each additional gigabyte used. Understanding how to evaluate piecewise functions helps in calculating costs, predicting outcomes, and making informed decisions in various scenarios.
To evaluate f ( t + 2 ) when t = 2 , we first calculate t + 2 = 4 . Then, since 4 ≥ 3 , we use the piecewise function f ( x ) = x 2 − 2 x to find that f ( 4 ) = 8 . Thus, the answer is 8 .
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