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}{lll}
x+2 & \text { if } & x\ \textless \ 1 \
x-2 & \text { if } & x \geq 1
\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: x + 2 for x < 1 and x − 2 for x ≥ 1 .
Graph the two parts of the function, noting the open and closed circles at x = 1 .
Identify the correct graph as option A.
Determine the range of the function by observing the graph: [ − 1 , ∞ ) .
Explanation
Understanding the Problem We are given a piecewise function and asked to graph it and determine its range. The function is defined as:
f ( x ) = { x + 2 x − 2 if if x < 1 x ≥ 1
We need to identify the correct graph and then find the range of the function based on the graph.
Analyzing the Graph The first part of the function, f ( x ) = x + 2 for x < 1 , is a line with a slope of 1 and a y-intercept of 2. Since x is strictly less than 1, the point at x = 1 is not included. When x = 1 , f ( 1 ) = 1 + 2 = 3 , so there is an open circle at the point ( 1 , 3 ) .
The second part of the function, f ( x ) = x − 2 for x ≥ 1 , is a line with a slope of 1 and a y-intercept of -2. Since x is greater than or equal to 1, the point at x = 1 is included. When x = 1 , f ( 1 ) = 1 − 2 = − 1 , so there is a closed circle at the point ( 1 , − 1 ) .
Looking at the options, graph A matches this description.
Determining the Range The graph consists of two parts:
For x < 1 , the function is f ( x ) = x + 2 . As x approaches 1 from the left, f ( x ) approaches 1 + 2 = 3 . So, the values of f ( x ) are less than 3.
For x ≥ 1 , the function is f ( x ) = x − 2 . At x = 1 , f ( 1 ) = 1 − 2 = − 1 . As x increases from 1, f ( x ) also increases. So, the values of f ( x ) are greater than or equal to -1.
Combining these two parts, the range of the function is [ − 1 , ∞ ) .
Final Answer The correct graph is A, and the range of the function is [ − 1 , ∞ ) .
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 one rate for the first 100 minutes and a different rate for each additional minute. Similarly, income tax brackets are a piecewise function, where the tax rate changes depending on the income level. Understanding piecewise functions helps in analyzing and predicting outcomes in such scenarios.
The correct graph is A, and the range of the function is [ − 1 , ∞ ) . The function combines a line increasing to the left and another line increasing to the right starting from point (1, -1).
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