Evaluate the step function for the given input values.
[tex]\begin{array}{l}
g(x)=\left\{\begin{array}{ll}
-4, & -3 \leq x\ \textless \ -1 \\
-1, & -1 \leq x\ \textless \ 2 \\
3, & 2 \leq x\ \textless \ 4 \\
5, & x \geq 4
\end{array}\right. \\
g(2)=\square \\
g(-2)=\square \\
g(5)=\square
\end{array}[/tex]
Answer (1)
Identify the intervals for the step function g ( x ) .
Determine that x = 2 falls in the interval 2 ≤ x < 4 , so g ( 2 ) = 3 .
Determine that x = − 2 falls in the interval − 3 ≤ x < − 1 , so g ( − 2 ) = − 4 .
Determine that x = 5 falls in the interval x ≥ 4 , so g ( 5 ) = 5 .
The final answer is: g ( 2 ) = 3 , g ( − 2 ) = − 4 , g ( 5 ) = 5 , so g ( 2 ) = 3 , g ( − 2 ) = − 4 , g ( 5 ) = 5
Explanation
Understanding the Problem We are given a step function g ( x ) and we need to evaluate it at x = 2 , x = − 2 , and x = 5 . A step function is a piecewise function that is constant on each interval of its domain. To evaluate g ( x ) at a specific value, we need to determine which interval contains that value and then use the corresponding constant value of the function on that interval.
Evaluating g(2) First, let's find g ( 2 ) . We need to find the interval that contains x = 2 . From the definition of g ( x ) , we see that the interval 2 ≤ x < 4 contains x = 2 . Therefore, g ( 2 ) = 3 .
Evaluating g(-2) Next, let's find g ( − 2 ) . We need to find the interval that contains x = − 2 . From the definition of g ( x ) , we see that the interval − 3 l e x < − 1 contains x = − 2 . Therefore, g ( − 2 ) = − 4 .
Evaluating g(5) Finally, let's find g ( 5 ) . We need to find the interval that contains x = 5 . From the definition of g ( x ) , we see that the interval x g e 4 contains x = 5 . Therefore, g ( 5 ) = 5 .
Final Answer Therefore, we have g ( 2 ) = 3 , g ( − 2 ) = − 4 , and g ( 5 ) = 5 .
Examples
Step functions are used in various real-world applications, such as modeling the cost of shipping based on weight intervals or calculating income tax based on income brackets. For example, a shipping company might charge a flat rate for packages weighing up to 1 kg, a higher rate for packages weighing between 1 kg and 5 kg, and so on. Similarly, income tax is often calculated using a step function, where different tax rates apply to different income brackets. Understanding step functions helps in analyzing and predicting outcomes in these scenarios.
