Given the piecewise function shown below, select all of the statements that are true.
[tex]f(x)=\left\{\begin{array}{c}2 x, x\ \textless \ 1 \\ 5, x=1 \\ x^2, x\ \textgreater \ 1\end{array}\right\[/tex]
A. [tex]f(1)=5[/tex]
B. [tex]f(2)=4[/tex]
C. [tex]f(-2)=4[/tex]
D. [tex]f(5)=1[/tex]
Answer (2)
Evaluate f ( 1 ) : Since x = 1 , f ( 1 ) = 5 , so statement A is true.
Evaluate f ( 2 ) : Since 1"> x = 2 > 1 , f ( 2 ) = 2 2 = 4 , so statement B is true.
Evaluate f ( − 2 ) : Since x = − 2 < 1 , f ( − 2 ) = 2 ( − 2 ) = − 4 , so statement C is false.
Evaluate f ( 5 ) : Since 1"> x = 5 > 1 , f ( 5 ) = 5 2 = 25 , so statement D is false.
The true statements are A and B.
Explanation
Understanding the Piecewise Function We are given a piecewise function and we need to evaluate the function at specific points to determine which statements are true. The piecewise function is defined as:
1 \end{array}\right\}"> f ( x ) = ⎩ ⎨ ⎧ 2 x , x < 1 5 , x = 1 x 2 , x > 1 ⎭ ⎬ ⎫
We will evaluate the function at x = 1 , 2 , − 2 , and 5 and compare the results with the given statements.
Evaluating f(1) Statement A: f ( 1 ) = 5 When x = 1 , the function is defined as f ( 1 ) = 5 . Therefore, statement A is true.
Evaluating f(2) Statement B: f ( 2 ) = 4 When x = 2 , since 1"> 2 > 1 , the function is defined as f ( x ) = x 2 . So, f ( 2 ) = 2 2 = 4 . Therefore, statement B is true.
Evaluating f(-2) Statement C: f ( − 2 ) = 4 When x = − 2 , since − 2 < 1 , the function is defined as f ( x ) = 2 x . So, f ( − 2 ) = 2 ∗ ( − 2 ) = − 4 . Therefore, statement C is false because f ( − 2 ) = − 4 , not 4 .
Evaluating f(5) Statement D: f ( 5 ) = 1 When x = 5 , since 1"> 5 > 1 , the function is defined as f ( x ) = x 2 . So, f ( 5 ) = 5 2 = 25 . Therefore, statement D is false because f ( 5 ) = 25 , not 1 .
Conclusion Based on our evaluations, statements A and B are true, while statements C and D are false.
Therefore, the correct statements are A and B.
Examples
Piecewise functions are used in real life to model situations where the rule or relationship changes based on the input. For example, income tax brackets are a piecewise function where the tax rate changes based on income. Another example is the cost of electricity, where the price per kilowatt-hour may change based on the amount of electricity used. Understanding how to evaluate piecewise functions is essential for analyzing these types of real-world scenarios.
The true statements are A and B. A evaluates to 5, and B evaluates to 4, while C evaluates to -4 and D evaluates to 25. Therefore, statements A and B are confirmed to be true based on the piecewise function definitions.
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