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Magazine Chapter 1: Problem 25
Graph the functions.
$$f(x)=\left\\{\begin{array}{ll}x, & 0 \leq x \leq 1 \\\2-x, & 1
Short Answer
Expert verified
The piecewise function consists of two linear parts: from (0,0) to (1,1) for \( f(x) = x \), and from slightly above (1,1) to (2,0) for \( f(x) = 2-x \).
Step by step solution
01
Identify the Pieces of the Function
The function is a piecewise function defined with two pieces. The first piece is \( f(x) = x \) for \( 0 \leq x \leq 1 \). The second piece is \( f(x) = 2 - x \) for \( 1 < x \leq 2 \). Each piece covers a specific interval of \(x\).
02
Graph the First Piece
For the interval \( 0 \leq x \leq 1 \), plot the function \( f(x) = x \). This is a straight line passing through the origin (0,0) and ending at the point (1,1). Make sure the point at \( x = 1 \) is solid, indicating it is included in this piece.
03
Graph the Second Piece
For the interval \( 1 < x \leq 2 \), plot the function \( f(x) = 2 - x \). This is a straight line starting from \( x = 1 \) towards the point (1,1), but takes effect only after \( x = 1 \). The line goes through (2,0) with a solid point at \( x = 2 \) indicating the endpoint is included.
04
Combine Graphs into One
Combine the two linear segments into one graph. Starting from (0,0), follow the first segment to (1,1) with a solid point, but note a hollow point at \( x = 1 \) for the stepping off of the second line, which officially begins just after \( x=1 \). Finally, continue to (2,0) with a solid point.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Functions
Graphing functions is an essential skill in mathematics that helps us visualize how a function behaves. It involves plotting points on a coordinate grid where each point represents an input-output pair, meaning for each value of \( x \), we find the corresponding value of \( f(x) \) and mark it on the graph.
To graph a function effectively, follow these steps:
To graph a function effectively, follow these steps:
- Identify the function type and its specific equation, checking for any restrictions or domains that it must adhere to.
- Create a table of values by choosing values of \( x \) within the domain and calculating corresponding values of \( f(x) \).
- Plot each point \((x, f(x))\) on the coordinate plane carefully, ensuring accuracy with each placement.
- Connect the points if necessary, remembering that some graphs like piecewise functions might not have smooth, connected lines but rather separate segments.
Linear Functions
Linear functions are one of the simplest and most fundamental types of functions in mathematics. They are characterized by the formula \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The graph of a linear function is a straight line.
Some key characteristics of linear functions are:
Some key characteristics of linear functions are:
- Constant rate of change: The slope \( m \) indicates the rate of change of the function. A positive slope means the function is increasing, while a negative slope indicates it's decreasing.
- Graphical representation: The line will cross the y-axis at the point \((0, b)\). All points on the line represent solutions to the equation \( y = mx + b \).
- Domain and Range: The domain is all real numbers, and the range is also all real numbers unless restricted by context.
Piecewise Function Graphing
Graphing piecewise functions involves plotting segments of different functions over specified intervals. This type of graphing is unique because the function has multiple expressions that apply to different intervals of the domain.
Here's how you can approach graphing a piecewise function:
Here's how you can approach graphing a piecewise function:
- Identify each piece: Carefully denote each part of the function, understanding what expression applies over what interval of \( x \). For example, the function \( f(x)=\{x,\ 0 \leq x \leq 1;\ 2-x,\ 1
- Graph segments separately: Begin plotting each segment on the graph, ensuring that you respect any endpoints (closed or open) as the function specifies. These might include solid points to show inclusion or hollow points to indicate that an endpoint is not included.
- Combine the segments: Once each piece is accurately graphed, combine them to form the entire piecewise function graph. Verify that transitions between segments align correctly in terms of continuity as defined by the function.
