Problem 39 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph consists of a horizontal line at \( y=3 \) for \( x<2 \), and a line with a slope of 1 starting from \( (2, 1) \) for \( x \geq 2 \).

Step 1: Understand the Function's Definition

The function is piecewise, meaning it's defined by different expressions depending on the value of \( x \). For \( x < 2 \), the function is constant, \( f(x) = 3 \). For \( x \geq 2 \), the function has a linear form \( f(x) = x - 1 \).

Step 2: Develop the Graph for \( x < 2 \)

The first part of the function, \( f(x) = 3 \), is a horizontal line. For values of \( x \) less than 2, plot a horizontal line at \( y = 3 \). This line should stop at \( x = 2 \) and not include the endpoint. Indicate an open circle at the point \( (2, 3) \) to show that this point is not included.

Step 3: Develop the Graph for \( x \geq 2 \)

The second part of the function is defined by \( f(x) = x - 1 \), which is a straight line with a slope of 1 and intercept of -1. For \( x \geq 2 \), the graph will include this linear part. Start plotting from \( x = 2 \, ( ext{inclusive}) \) with the value \( f(2) = 1 \), and continue with the slope of 1.

Step 4: Combine Both Parts

Ensure both sections are correctly drawn with their specific boundaries: the line at \( y = 3 \) for \( x < 2 \) with an open circle at \( (2, 3) \) and the linear parts for \( x \geq 2 \) starting at \( (2, 1) \). The graph is continuous but has a jump at \( x = 2 \).

Key Concepts

Graphing Piecewise Functions

Graphing piecewise functions involves dealing with multiple expressions or rules within the same function. Each sub-function has its own domain or range over which it is applicable. In our exercise, we deal with two parts of the function, each corresponding to a different range of values for \(x\). The first part, applicable for \(x < 2\), is a constant function \(f(x) = 3\). This means the graph is a horizontal line. In these types of functions, plot the line across the extended part of the graph until the boundary at \(x = 2\), then indicate that this end is not included by using an open circle.

The second part of the function comes into play for \(x \geq 2\). This is a linear function \(f(x) = x - 1\), which you plot starting right at \(x = 2\) including this point, as indicated by a filled circle. Piecewise functions can appear intimidating but breaking them into their pieces, just like playing with building blocks, simplifies the process.

Continuous and Discontinuous Functions

Functions are considered continuous if you can draw them without lifting your pencil from the paper. For continuous and discontinuous functions, understanding the transition points and any potential gaps are crucial. In the provided exercise, the piecewise function has a discontinuity at \(x = 2\), which is represented as a 'jump' in the graph.

The key principle of continuity in mathematics considers whether you can connect two points smoothly. Here, the value of the function changes abruptly from \(y = 3\) to \(y = 1\) at \(x = 2\). Because of this jump, the graph is not continuous at this specific point. Understanding these concepts is vital for identifying how a function behaves across its domain.

Plotting Linear Functions

The task of plotting linear functions is foundational in mathematics. To understand how to plot them, one must recognize the formula \(y = mx + b\), where \(m\) is the slope and \(b\) the y-intercept. In our function, the linear part for \(x \geq 2\) is \(f(x) = x - 1\). This corresponds to a slope \(m\) of 1 and a y-intercept \(b\) of -1.

A slope of 1 means that for every unit increase in \(x\), \(y\) also increases by 1 unit. Begin plotting at \(x = 2\), where \(f(2) = 1\), and continue with a consistent slope. This methodical approach makes plotting linear sections of piecewise functions straightforward. Practice with linear functions enhances understanding and prepares one for tackling more complex direct or inverse relationships in functions.