The domain of the piecewise function is $(-\infty, \infty)$.
a. Graph the function.
b. Use your graph to determine the function's range.
[tex]f(x)=\left\{\begin{array}{rll} 6 x & if & x \leq 0 \\ 6 & if & x \ \textgreater \ 0 \end{array}\right.[/tex]
a. Choose the correct graph below.
A.
B.
C.
D.
Answer (2)
The function is a piecewise function with two parts: f ( x ) = 6 x for x ≤ 0 and f ( x ) = 6 for 0"> x > 0 .
The graph consists of a line with slope 6 for x ≤ 0 and a horizontal line at y = 6 for 0"> x > 0 .
The range is determined by considering the possible y-values for each part of the function.
The range of the function is ( − ∞ , 0 ] ∪ { 6 } .
Explanation
Understanding the Problem We are given a piecewise function and asked to graph it and determine its range. The function is defined as: 0 \end{array}\right."> f ( x ) = { 6 x 6 if if x ≤ 0 x > 0 We need to identify the correct graph and then find the range of the function.
Analyzing the Function First, let's analyze the two parts of the piecewise function:
For x ≤ 0 , the function is f ( x ) = 6 x . This is a linear function with a slope of 6, passing through the origin (0,0). Since x is less than or equal to 0, this part of the graph exists for all non-positive x values, including x = 0 .
For 0"> x > 0 , the function is f ( x ) = 6 . This is a horizontal line at y = 6 . Since x is strictly greater than 0, this part of the graph exists for all positive x values, but not at x = 0 . This means there will be an open circle at (0,6).
Identifying the Correct Graph Based on the analysis, we can conclude that the correct graph should have a line with a slope of 6 for x ≤ 0 and a horizontal line at y = 6 for 0"> x > 0 . Looking at the options, graph A matches this description.
Determining the Range Now, let's determine the range of the function. The range is the set of all possible y values that the function can take.
For x ≤ 0 , f ( x ) = 6 x can take any value from − ∞ to 0, inclusive. So, the interval ( − ∞ , 0 ] is part of the range.
For 0"> x > 0 , f ( x ) = 6 . The function only takes the value 6. So, we include {6} in the range. Combining these, the range of the function is ( − ∞ , 0 ] ∪ { 6 } .
Final Answer Therefore, the correct graph is A, and the range of the function is ( − ∞ , 0 ] ∪ { 6 } .
Examples
Piecewise functions are used in real life to model situations where the rule for a function changes depending on the input value. For example, a cell phone plan might charge a different rate for data usage depending on whether you are below or above a certain data limit. Similarly, income tax brackets are a piecewise function where the tax rate changes based on your income level. Understanding piecewise functions helps us analyze and predict outcomes in these scenarios.
The graph of the piecewise function features a line with a slope of 6 for x ≤ 0 and a horizontal line at y = 6 for 0. The range of the function is"> x > 0. T h er an g eo f t h e f u n c t i o ni s (-\infty, 0] \cup {6}$. Based on these criteria, choose the appropriate option from the provided graphs.
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