Which graph represents the piecewise-defined function $f(x)=\left\{\begin{array}{ll}-1.5 x+3.5, & x<2 \\ 4+x, & x \geq 2\end{array}\right.?$
Answer (2)
The function is a piecewise-defined function with two linear pieces.
For x < 2 , f ( x ) = − 1.5 x + 3.5 , which is a line with slope -1.5 and y-intercept 3.5, approaching the point (2, 0.5) with an open circle.
For x ≥ 2 , f ( x ) = 4 + x , which is a line with slope 1 and y-intercept 4, starting at the point (2, 6) with a closed circle.
The graph that matches these characteristics represents the piecewise-defined function. The correct graph has a line approaching ( 2 , 0.5 ) from the left with an open circle and a line starting at ( 2 , 6 ) with a closed circle. Therefore, the answer is the graph that matches these characteristics.
Explanation
Understanding the Piecewise Function We are given a piecewise-defined function and asked to identify its graph. The function is defined as: f ( x ) = { − 1.5 x + 3.5 , 4 + x , x < 2 x ≥ 2
This means that for values of x less than 2, the function behaves like the linear function − 1.5 x + 3.5 . For values of x greater than or equal to 2, the function behaves like the linear function 4 + x . We need to pay close attention to what happens at x = 2 , since that's where the function definition changes.
Analyzing the First Piece Let's analyze the first piece of the function, f ( x ) = − 1.5 x + 3.5 for x < 2 . This is a linear function with a slope of − 1.5 and a y-intercept of 3.5 . Since this piece is defined for x < 2 , we need to evaluate the function at x = 2 to see where the graph approaches. f ( 2 ) = − 1.5 ( 2 ) + 3.5 = − 3 + 3.5 = 0.5 So, as x approaches 2 from the left, f ( x ) approaches 0.5. Since the domain is x < 2 , we will have an open circle at the point ( 2 , 0.5 ) .
Analyzing the Second Piece Now let's analyze the second piece of the function, f ( x ) = 4 + x for x ≥ 2 . This is a linear function with a slope of 1 and a y-intercept of 4 . Since this piece is defined for x ≥ 2 , we need to evaluate the function at x = 2 to see where the graph starts. f ( 2 ) = 4 + 2 = 6 So, at x = 2 , f ( x ) = 6 . Since the domain is x ≥ 2 , we will have a closed circle at the point ( 2 , 6 ) .
Identifying the Graph In summary, the graph of the piecewise function will have two parts:
For x < 2 , it will look like a line with a slope of − 1.5 and a y-intercept of 3.5 , approaching the point ( 2 , 0.5 ) with an open circle.
For x ≥ 2 , it will look like a line with a slope of 1 and a y-intercept of 4 , starting at the point ( 2 , 6 ) with a closed circle.
Therefore, we need to look for a graph that matches these characteristics.
Examples
Piecewise functions are used in real life to model situations where the rule or relationship changes based on the input value. For example, cell phone plans often have different rates 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 types of scenarios.
The piecewise-defined function consists of two linear parts: one approaches ( 2 , 0.5 ) with an open circle for x < 2 , and the other starts from ( 2 , 6 ) with a closed circle for x ≥ 2 . The graph should reflect these characteristics accordingly. The correct graph will showcase these features at the designated point of transition at x = 2 .
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