Which graph is the graph of this function?
[tex]f(x)=\left\{\begin{array}{cl}3 \sqrt{x+1} & \text { if } 0 \leq x\ \textless \ 3 \\ 5-x & \text { if } 3 \leq x \leq 5\end{array}\right.[/tex]
A. graph A
B. graph B
C. graph C
D. graph D
Answer (2)
The piecewise function is defined in two segments: one from 0 to 3 , which rises from ( 0 , 3 ) → ( 3 , 6 ) and another from 3 to 5 , descending from ( 3 , 2 ) → ( 5 , 0 ) . The correct graph for this function is option B.
;
The function is defined piecewise: f ( x ) = 3 x + 1 for 0 ≤ x < 3 and f ( x ) = 5 − x for 3 ≤ x ≤ 5 .
The first part starts at ( 0 , 3 ) and approaches ( 3 , 6 ) .
The second part starts at ( 3 , 2 ) and ends at ( 5 , 0 ) .
Therefore, the correct graph is B. B
Explanation
Understanding the Piecewise Function We are given a piecewise function: f ( x ) = { 3 x + 1 5 − x if 0 ≤ x < 3 if 3 ≤ x ≤ 5 Our goal is to identify the correct graph of this function from the given options.
Analyzing the First Part of the Function First, let's analyze the first part of the function, f ( x ) = 3 x + 1 for 0 ≤ x < 3 .
When x = 0 , f ( 0 ) = 3 0 + 1 = 3 1 = 3 . So, the graph starts at the point ( 0 , 3 ) .
As x approaches 3 (but not including 3 ), f ( x ) approaches 3 3 + 1 = 3 4 = 3 × 2 = 6 . So, the graph approaches the point ( 3 , 6 ) but does not reach it.
Analyzing the Second Part of the Function Now, let's analyze the second part of the function, f ( x ) = 5 − x for 3 ≤ x ≤ 5 .
When x = 3 , f ( 3 ) = 5 − 3 = 2 . So, the graph starts at the point ( 3 , 2 ) .
When x = 5 , f ( 5 ) = 5 − 5 = 0 . So, the graph ends at the point ( 5 , 0 ) .
Identifying the Correct Graph Let's summarize the key points:
The first part of the function is f ( x ) = 3 x + 1 for 0 ≤ x < 3 , which starts at ( 0 , 3 ) and approaches ( 3 , 6 ) .
The second part of the function is f ( x ) = 5 − x for 3 ≤ x ≤ 5 , which starts at ( 3 , 2 ) and ends at ( 5 , 0 ) .
Based on these points, we can analyze the given graphs to find the correct one. Graph B matches these characteristics.
Examples
Piecewise functions are used in real life to model situations where different rules or conditions apply over different intervals. For example, a cell phone plan might charge one rate for the first 100 minutes and a different rate for each additional minute. Similarly, income tax brackets are defined using a piecewise function, where the tax rate changes as income increases. Understanding piecewise functions helps in analyzing and predicting outcomes in these scenarios.
