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Chapter 2: Problem 39

Sketch the graph of the piecewise defined function. $$f(x)=\left\\{\begin{array}{ll}3 & \text { if } x<2 \\ x-1 & \text { if } x \geq 2\end{array}\right.$$

Short Answer

Expert verified
Graph a horizontal line at \( y = 3 \) for \( x < 2 \) and a line \( y = x-1 \) starting at \( (2, 1) \) for \( x \geq 2 \).

Step by step solution

01

Understand the Function Definition

The function \( f(x) \) is defined in two different ways depending on the value of \( x \). There are two cases: if \( x < 2 \) then \( f(x) = 3 \) and if \( x \geq 2 \) then \( f(x) = x - 1 \). Recognize that this is a piecewise function with two pieces: a constant function and a linear function.
02

Graph the First Piece

For the first piece, where \( x < 2 \), the function \( f(x) = 3 \) is a horizontal line. On a graph, draw a horizontal line at \( y = 3 \) for all \( x < 2 \). Since the domain is only for \( x < 2 \), plot a solid dot at \( (2, 3) \) to indicate the endpoint of this segment of the function.
03

Graph the Second Piece

For the second piece, where \( x \geq 2 \), the function \( f(x) = x - 1 \) is a linear function with a slope of \( 1 \) and y-intercept of \( -1 \). Begin by plotting the point at \( (2, 1) \) because when \( x = 2 \), \( f(x) = 1 \). Since this piece is defined for \( x \geq 2 \), draw a solid line starting at \( (2, 1) \) extending to the right.
04

Verify the Transition at x=2

Observe the transition between the two pieces at \( x = 2 \). The first piece ends at \( (2, 3) \) but indicates a value less than 2. The second piece starts at \( (2, 1) \) continuing for \( x \geq 2 \). Note that \( x = 2 \) is covered by the second piece only, making it continuous on this point.
05

Sketch the Full Graph

Combine the two pieces on a single graph: a horizontal line \( y = 3 \) for \( x < 2 \), and a line starting from \( (2, 1) \) going upwards with a slope of \( 1 \) for \( x \geq 2 \). Ensure the graph is coherent with the conditions of each piece, showing a clear, clean transition at \( x = 2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Piecewise Functions
Piecewise functions are a type of function that have different expressions based on the input values. In this context, the function \(f(x)\) is defined differently for \(x < 2\) and \(x \geq 2\). This creates two distinct "pieces" of the graph, each with their own characteristics. To graph a piecewise function effectively:
  • Identify the different pieces and their corresponding conditions.
  • Graph each piece separately over its specific domain.
  • Check the transition points to ensure smooth transitions or note where discontinuities occur.
By following these steps, you can create a complete and accurate representation of the function.
Constant Function
A constant function is a function that always returns the same value, regardless of the input. In the context of the given piecewise function, the piece defined by \(f(x) = 3\) for \(x < 2\) is a constant function. This means:
  • The graph will be a horizontal line.
  • All points on this line will have the same \(y\)-coordinate, which is \(3\).
  • In this particular function, the graph is valid for all \(x\) less than 2.
To represent it on the graph, draw a horizontal line at \(y=3\) extending to the left as x approaches the transition point of \(2\).
Linear Function
In the piecewise function, the expression \(f(x) = x - 1\) for \(x \geq 2\) defines a linear function. A linear function has the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here:
  • The slope \(m = 1\) tells us that for every unit increase in \(x\), \(y\) increases by 1.
  • The y-intercept is \(-1\), which means the line would cross the y-axis at \(y = -1\) if extended.
To graph this part, start at the point \((2, 1)\) 鈥� the endpoint of the previous piece and the starting point for this linear expression. Extend a line with a slope of 1 to the right, covering the domain \(x \geq 2\).
Transition Point
The transition point in a piecewise function is where the function's definition changes from one expression to another. In this case, \(x = 2\) is the transition point, where the function shifts from being a constant \(f(x) = 3\) to a linear expression \(f(x) = x - 1\). At transition points:
  • Check if the values of the function match from both sides.
  • Determine whether the graph needs to be continuous or if a break occurs.
For this function, while the constant line ends at \((2, 3)\), the linear line starts at \((2, 1)\). Thus, at \(x = 2\), the function value in the linear piece applies, giving \(f(2) = 1\). The graph will show a downward step from the constant to the linear segment at this point, indicating a clear transition.

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Most popular questions from this chapter

A car dealership advertises a \(15 \%\) discount on all its new cars. In addition, the manufacturer offers a \(\$ 1000\) rebate on the purchase of a new car. Let \(x\) represent the sticker price of the car. (a) Suppose only the \(15 \%\) discount applies. Find a function \(f\) that models the purchase price of the car as a function of the sticker price \(x\) (b) Suppose only the \(\$ 1000\) rebate applies. Find a function \(g\) that models the purchase price of the car as a function of the sticker price \(x\) (c) Find a formula for \(H=f \circ g\) (d) Find \(H^{-1} .\) What does \(H^{-1}\) represent? (e) Find \(H^{-1}(13,000) .\) What does your answer represent?

Express the function in the form \(f \circ g\) $$F(x)=(x-9)^{5}$$

Find a function whose graph is the given curve. The line segment joining the points (-3,-2) and (6,3)

At the beginning of this section we discussed three examples of everyday, ordinary functions: Height is a function of age, temperature is a function of date, and postage cost is a function of weight. Give three other examples of functions from everyday life.

Find \(f \circ g \circ h\) $$f(x)=\sqrt{x}, \quad g(x)=\frac{x}{x-1}, \quad h(x)=\sqrt[3]{x}$$
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