Given the piecewise function shown below, select all of the statements that are true.
[tex]f(x)=\left\{\begin{array}{c}-3 x, x\ \textless \ 0 \\4, x=0 \\x^2, x\ \textgreater \ 0\end{array}\right\}[/tex]
A. [tex]f(2)=4[/tex]
B. [tex]f(77)=-3[/tex]
C. [tex]f(4)=0[/tex]
D. [tex]f(3)=9[/tex]
Answer (1)
Evaluate f ( 2 ) : Since 0"> 2 > 0 , f ( 2 ) = 2 2 = 4 , so statement A is true.
Evaluate f ( 7 ) and f ( − 7 ) : f ( 7 ) = 7 2 = 49 and f ( − 7 ) = − 3 ( − 7 ) = 21 , so statement B is false.
Evaluate f ( 4 ) : Since 0"> 4 > 0 , f ( 4 ) = 4 2 = 16 , so statement C is false.
Evaluate f ( 3 ) : Since 0"> 3 > 0 , f ( 3 ) = 3 2 = 9 , so statement D is true.
The true statements are A and D. A , D
Explanation
Analyzing the Piecewise Function We are given a piecewise function and asked to determine which of the given statements are true. Let's analyze each statement individually. The piecewise function is defined as:
0 \end{array}\right"> f(x)=\left\{\begin{array}{c} -3 x, x<0 \ 4, x=0 \ x^2, x>0 \end{array}\right
Evaluating Statement A Statement A: f ( 2 ) = 4 Since 0"> 2 > 0 , we use the third piece of the function, f ( x ) = x 2 . Thus, f ( 2 ) = 2 2 = 4 . So, statement A is true.
Evaluating Statement B Statement B: ( 77 ) = − 3 There seems to be a typo in this statement. Assuming it is f ( 7 ) = − 3 or f ( − 7 ) = − 3 , let's evaluate f ( 7 ) and f ( − 7 ) .
Since 0"> 7 > 0 , we use the third piece of the function, f ( x ) = x 2 . Thus, f ( 7 ) = 7 2 = 49 .
Since − 7 < 0 , we use the first piece of the function, f ( x ) = − 3 x . Thus, f ( − 7 ) = − 3 ( − 7 ) = 21 .
Neither f ( 7 ) nor f ( − 7 ) equals − 3 . So, statement B is false.
Evaluating Statement C Statement C: f ( 4 ) = 0 Since 0"> 4 > 0 , we use the third piece of the function, f ( x ) = x 2 . Thus, f ( 4 ) = 4 2 = 16 . So, statement C is false.
Evaluating Statement D Statement D: f ( 3 ) = 9 Since 0"> 3 > 0 , we use the third piece of the function, f ( x ) = x 2 . Thus, f ( 3 ) = 3 2 = 9 . So, statement D is true.
Conclusion Therefore, the true statements are A and D.
Examples
Piecewise functions are used in real life to model situations where the rule or relationship changes based on the input. For example, cell phone plans often have different rates for data usage depending on how much data you've used in a month. The cost might be one rate for the first few gigabytes, and then a higher rate for any additional data. Similarly, income tax brackets are a form of piecewise function, where the tax rate changes depending on your income level. Understanding piecewise functions helps in analyzing and predicting outcomes in these scenarios.
