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)
To determine which graph corresponds to the given piecewise function f ( x ) , we need to consider how the function is defined on different intervals of x .
The function f ( x ) is defined as follows:
f ( x ) = 3 x + 1 for 0 ≤ x < 3
This part of the function represents a square root graph that starts at x = 0 and ends just before x = 3 .
The term x + 1 shifts the square root function one unit to the left, so it starts at 1 when x = 0 , resulting in f ( x ) = 3 ⋅ 1 = 3 .
At x = 3 , f ( x ) = 3 4 = 3 ⋅ 2 = 6 , but since x = 3 is not included in this part of the piecewise function, the open circle will appear at ( 3 , 6 ) in the graph.
f ( x ) = 5 − x for 3 ≤ x ≤ 5
This part of the function is a linear equation with a negative slope of -1, starting at x = 3 and ending at x = 5 .
At x = 3 , f ( x ) = 5 − 3 = 2 .
At x = 5 , f ( x ) = 5 − 5 = 0 .
Now, putting these details together:
For x from 0 to just under 3, the graph will show part of a square root curve starting at ( 0 , 3 ) and approaching ( 3 , 6 ) with an open circle.
For x from 3 to 5, the graph will show a straight line segment starting at ( 3 , 2 ) (a closed circle) and going to ( 5 , 0 ) (a closed circle).
By examining each provided graph against these criteria, you can determine which one matches. Since I don't have access to the visual graphs, please compare this description with options A, B, C, and D to find the correct graph.
