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 (1)
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 of the graph starts at ( 0 , 3 ) and approaches ( 3 , 6 ) with a square root curve.
The second part of the graph is a straight line segment from ( 3 , 2 ) to ( 5 , 0 ) .
By comparing these features to the given graphs, the correct graph can be identified. Without the graphs, a definitive answer is not possible.
Explanation
Analyze the piecewise function Let's analyze the given piecewise function to determine which graph represents it correctly. The function is defined as:
f ( x ) = { 3 x + 1 5 − x if 0 ≤ x < 3 if 3 ≤ x ≤ 5
We will analyze each part of the function separately.
Analyze the first part of the function First, consider the interval 0 ≤ x < 3 . The function is f ( x ) = 3 x + 1 .
At x = 0 , we have f ( 0 ) = 3 0 + 1 = 3 1 = 3 . So the graph starts at the point ( 0 , 3 ) .
As x approaches 3 (but not including x = 3 ), we have f ( x ) approaching 3 3 + 1 = 3 4 = 3 ( 2 ) = 6 . So the graph approaches the point ( 3 , 6 ) but does not include it.
Analyze the second part of the function Next, consider the interval 3 ≤ x ≤ 5 . The function is f ( x ) = 5 − x .
At x = 3 , we have f ( 3 ) = 5 − 3 = 2 . So the graph starts at the point ( 3 , 2 ) .
At x = 5 , we have f ( 5 ) = 5 − 5 = 0 . So the graph ends at the point ( 5 , 0 ) .
This part of the function is a straight line segment.
Summarize the features of the graph Now, let's summarize the key features of the graph:
For 0 ≤ x < 3 , the graph starts at ( 0 , 3 ) and approaches ( 3 , 6 ) with a square root curve.
For 3 ≤ x ≤ 5 , the graph is a straight line segment from ( 3 , 2 ) to ( 5 , 0 ) .
Based on these features, we can determine which of the given graphs matches this description. Without the graphs, I can't definitively pick one. However, I can describe what to look for:
The graph should start at ( 0 , 3 ) .
The graph should have a square root shape that approaches but does not reach ( 3 , 6 ) .
At x = 3 , the graph should jump down to ( 3 , 2 ) .
The graph should have a straight line from ( 3 , 2 ) to ( 5 , 0 ) .
Conclusion Without the actual graphs (A, B, C, and D), I can't give a definitive answer. However, by carefully comparing the features we've identified to each graph, you can determine the correct one.
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 of calls and a different rate for each additional minute. Similarly, the cost of electricity might vary depending on the time of day, with peak hours having higher rates than off-peak hours. Understanding piecewise functions helps us analyze and predict outcomes in these types of scenarios.
