The function [tex]g(x)[/tex] is defined as shown.
[tex]g(x)=\left\{\begin{array}{cc}
x-1, & -2 \leq x\ \textless \ -1 \
2 x+3, & -1 \leq x\ \textless \ 3 \
6-x, & x \geq 3
\end{array}\right.[/tex]
What is the value of [tex]g(3)[/tex] ?
A. 2
B. 3
C. 9
D. 14
Answer (2)
Determine the interval that contains x = 3 .
Since 3 ≥ 3 , use the third case: g ( x ) = 6 − x .
Substitute x = 3 into g ( x ) = 6 − x .
Calculate g ( 3 ) = 6 − 3 = 3 , so the final answer is 3 .
Explanation
Identifying the Correct Piecewise Function We are given a piecewise function g ( x ) and we want to find the value of g ( 3 ) . The function is defined as:
g ( x ) = ⎩ ⎨ ⎧ x − 1 , 2 x + 3 , 6 − x , − 2 ≤ x < − 1 − 1 ≤ x < 3 x ≥ 3
We need to determine which piece of the function applies when x = 3 .
Applying the Condition Since x = 3 , we look for the interval that includes 3 . We see that the condition x ≥ 3 applies. Therefore, we use the third piece of the function, which is g ( x ) = 6 − x .
Calculating g(3) Now we substitute x = 3 into the expression g ( x ) = 6 − x :
g ( 3 ) = 6 − 3 = 3
Final Answer Therefore, the value of g ( 3 ) is 3 .
Examples
Piecewise functions are used in real life to model situations where different rules or conditions apply over different intervals. 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 per gigabyte might be lower up to a certain threshold, and then increase if you exceed that threshold. Similarly, income tax brackets are a real-world application of piecewise functions, where different tax rates apply to different income ranges. Understanding piecewise functions helps in analyzing and predicting outcomes in these scenarios.
The value of g ( 3 ) is calculated using the relevant expression for x ≥ 3 , which leads to g ( 3 ) = 6 − 3 = 3 . Therefore, the correct answer is option B: 3.
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