Problem 28 - Thomas Calculus

Question

Graph the functions in Exercises \(25-28\) $$ G(x)=\left\\{\begin{array}{ll}{1 / x,} & {x<0} \\ {x,} & {0 \leq x}\end{array}\right. $$

Step-by-Step Solution

Answer

The graph is a hyperbola for \( x < 0 \) and a line for \( x \geq 0 \), with a discontinuity at \( x = 0 \).

Step 1: Understanding the Function

The given function G(x) is a piecewise function, which is defined differently for different intervals of x. It has two parts: when x is less than 0, the function is defined as \( G(x) = \frac{1}{x} \), and when x is greater than or equal to 0, the function is defined as \( G(x) = x \).

Step 2: Plotting Part 1: \( G(x) = \frac{1}{x} \) for \( x < 0 \)

For the interval \( x < 0 \), the function follows \( G(x) = \frac{1}{x} \). This portion of the graph is a hyperbola in the second quadrant. Key points to plot include \((x, G(x)) = (-1, -1)\) and \((-2, -0.5)\). The curve approaches infinity as \( x \) approaches 0 from the left.

Step 3: Plotting Part 2: \( G(x) = x \) for \( x \geq 0 \)

For the interval \( x \geq 0 \), the function is \( G(x) = x \). This is a straight line passing through the origin with a slope of 1. Key points to plot include \((0, 0)\), \((1, 1)\), \((2, 2)\), etc. The line continues infinitely in the positive direction.

Step 4: Checking Discontinuity at \( x = 0 \)

At \( x = 0 \), there is a transition in the function from \( \frac{1}{x} \) to \( x \). Evaluate the left-hand limit as \( x \to 0^- \), which tends to \(-\infty\), and the right-hand limit as \( x \to 0^+ \), which equals 0. Since the two limits are not equal, there is a discontinuity at \( x = 0 \).

Key Concepts

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize how a function behaves. A piecewise function, like the one given in this exercise, is defined by different expressions depending on the value of the independent variable (in this case, x). Understanding the behavior of each part of the function is crucial for accurate graphing.

Here are some steps to graph a piecewise function like \( G(x) \) effectively:

• Identify each segment of the piecewise function and the intervals over which they are defined.
• For each interval, calculate some key points. Plotting these points will give you a shape of the graph.
• Consider the behavior of the graph as x approaches the boundaries of the intervals.

For \( G(x) \), you plot a hyperbola in the second quadrant for \( x < 0 \) and a straight line for \( x \geq 0 \). Focus on points like \((-1, -1)\) and \((0,0)\) to get the overall shape. Fill in more points for accuracy and connect them smoothly.

Discontinuity

Discontinuity refers to a point at which a function is not continuous, meaning there is an abrupt change or gap in the graph. For piecewise functions, discontinuities often occur at the boundaries where one piece of the function changes to another. Understanding discontinuity is essential for accurately interpreting and graphing functions.

In the function \( G(x) \), discontinuity occurs at \( x = 0 \). This is because as \( x \) approaches 0 from the left \((x < 0)\), the function approaches \(-\infty\) due to the \( \frac{1}{x} \) part. On the other hand, as \( x \) approaches 0 from the right \((x \geq 0)\), the function equals 0 as per the \( x \) section. Since these two limits are different, \( x = 0 \) is not continuous. Graphically, this results in a break or jump at \( x = 0 \).

Function Intervals

Function intervals describe the specific ranges of x-values over which parts of a function are defined differently. Understanding these intervals is vital when working with piecewise functions, as each piece operates under different rules in its respective interval.

For the piecewise function \( G(x) \) given in the exercise, there are two intervals:

• \( x < 0 \): Here, the function is governed by \( G(x) = \frac{1}{x} \), which forms a hyperbola.
• \( x \geq 0 \): Here, the function follows \( G(x) = x \), creating a linear relationship.

When graphing, acknowledging where each interval starts and ends, and plotting them separately, ensures a clear and accurate graph. Always be aware of the function's behavior at the boundary points, as this can affect both the continuity and the overall appearance of the graph.