Which graph is the graph of this function?
[tex]f(x)=\left\{\begin{array}{ll}\frac{1}{2} x-2 & \text { if } x \leq 6 \\ -x-1 & \text { if } x\ \textgreater \ 6\end{array}\right.[/tex]
A. graph A
B. graph B
C. graph C
D. graph D
Answer (2)
The function is a piecewise function with two linear segments.
For x ≤ 6 , f ( x ) = 2 1 x − 2 , a line with slope 2 1 and passing through ( 6 , 1 ) .
For 6"> x > 6 , f ( x ) = − x − 1 , a line with slope -1.
The graph has a break at x = 6 . Without the images of the graphs, it is impossible to choose the correct option, but the key features that the correct graph must possess are described. The final answer is that we need to see the graphs to answer the question.
Explanation
Analyze the Piecewise Function The problem asks us to identify the correct graph for the given piecewise function:
6 \end{array}\right."> f ( x ) = { 2 1 x − 2 − x − 1 if x ≤ 6 if x > 6
We need to analyze the function and compare its characteristics with the provided graphs.
Analyze the First Part of the Function Let's examine the first part of the function, which is defined for x ≤ 6 :
f ( x ) = 2 1 x − 2
This is a linear function with a slope of 2 1 and a y-intercept of -2. At x = 6 , the value of the function is:
f ( 6 ) = 2 1 ( 6 ) − 2 = 3 − 2 = 1
So, the graph includes the point ( 6 , 1 ) .
Analyze the Second Part of the Function Now let's look at the second part of the function, which is defined for 6"> x > 6 :
f ( x ) = − x − 1
This is a linear function with a slope of -1 and a y-intercept of -1. This part of the function is not defined at x = 6 , but if it were, its value would be:
f ( 6 ) = − 6 − 1 = − 7
Since this part is defined for 6"> x > 6 , the graph approaches the point ( 6 , − 7 ) but does not include it (it's an open point).
Summarize the Characteristics of the Graph Based on our analysis:
For x ≤ 6 , the function is f ( x ) = 2 1 x − 2 , which is a line with a slope of 2 1 and passes through the point ( 6 , 1 ) .
For 6"> x > 6 , the function is f ( x ) = − x − 1 , which is a line with a slope of -1. The graph has a break at x = 6 , and the function approaches ( 6 , − 7 ) but does not include it.
Without seeing the graphs, we can describe the correct graph. It should have two linear segments. The first segment should have a slope of 1/2 and end at the point (6,1) (inclusive). The second segment should have a slope of -1 and start just to the right of x=6.
Further Verification with Specific Points By evaluating the function at specific points, we can further confirm the graph's behavior. For example:
At x = 0 , f ( 0 ) = 2 1 ( 0 ) − 2 = − 2 . So the graph passes through ( 0 , − 2 ) .
At x = 7 , f ( 7 ) = − 7 − 1 = − 8 . So the graph passes through ( 7 , − 8 ) .
At x = − 2 , f ( − 2 ) = 2 1 ( − 2 ) − 2 = − 1 − 2 = − 3 . So the graph passes through ( − 2 , − 3 ) .
Therefore, without the images of the graphs, it is impossible to choose the correct option. However, we have described the key features that the correct graph must possess.
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 certain number of gigabytes of data and a different rate for additional data usage. Similarly, income tax brackets are defined using a piecewise function, where different tax rates apply to different income ranges. Understanding piecewise functions helps in analyzing and predicting outcomes in these scenarios.
To find the correct graph of the piecewise function, we analyzed the function's segments. The first part has a slope of 2 1 and ends at (6,1), while the second part has a slope of -1 starting just after x = 6 and approaches a value of -7. Without the visual graphs, we cannot choose an option, but key features of the graph are defined above.
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