Problem 30 - Calculus Early Transcendentals

Question

Sketching graphs of functions Sketch the graph of a function with the given properties. You do not need to find a formula for the function. $$\begin{aligned} &h(-1)=2, \lim _{x \rightarrow-1^{-}} h(x)=0, \lim _{x \rightarrow-1^{+}} h(x)=3\\\ &h(1)=\lim _{x \rightarrow 1^{-}} h(x)=1, \lim _{x \rightarrow 1^{+}} h(x)=4 \end{aligned}$$

Step-by-Step Solution

Answer

Short Answer: The graph of the function \(h(x)\) has a point at \((-1, 2)\) and a jump discontinuity at \(x = -1\) with left limit 0 and right limit 3. There is a point at \((1, 1)\), and the graph smoothly connects to this point from the left, although there is a jump discontinuity at \(x = 1\) with a limit from the right of 4.

Step 1: Understand the given properties

First, analyze each given property in the exercise: 1. \(h(-1) = 2\): This means that the function \(h(x)\) has a value of 2 when \(x = -1\). 2. \(\lim _{x \rightarrow-1^{-}} h(x) = 0\): This means that as \(x\) approaches \(-1\) from the left, the value of \(h(x)\) approaches 0. 3. \(\lim _{x \rightarrow-1^{+}} h(x) = 3\): This means that as \(x\) approaches \(-1\) from the right, the value of \(h(x)\) approaches 3. 4. \(h(1) = \lim _{x \rightarrow 1^{-}} h(x) = 1\): This means that the left limit of the function at \(x = 1\) is 1, and the function value at this point is also 1. 5. \(\lim _{x \rightarrow 1^{+}} h(x) = 4\): This means that as \(x\) approaches \(1\) from the right, the value of \(h(x)\) approaches 4.

Step 2: Sketch the graph using the properties

Here is a step-by-step guide to sketching the graph of the function using the given properties: 1. Mark the point \((-1, 2)\) on the graph, since this is the value of the function at \(x = -1\). 2. As \(x\) approaches \(-1\) from the left, the value of \(h(x)\) approaches 0. Draw a curve that approaches the point \((-1, 0)\) but does not touch it. 3. As \(x\) approaches \(-1\) from the right, the value of \(h(x)\) approaches 3. Draw a curve that approaches the point \((-1, 3)\) but does not touch it. There will be a jump discontinuity at \(x = -1\). 4. Mark the point \((1, 1)\) on the graph, since this is the value of the function at \(x = 1\). 5. As \(x\) approaches \(1\) from the left, the value of \(h(x)\) also approaches 1. This means that the graph must smoothly join the point \((1, 1)\) coming from the left. 6. As \(x\) approaches \(1\) from the right, the value of \(h(x)\) approaches 4. Draw a curve that approaches the point \((1, 4)\) but does not touch it. There will be a jump discontinuity at \(x = 1\). 7. Connect the segments of the graph as needed, ensuring that the properties are satisfied. The graph must show the following key features: 1. A point at \((-1, 2)\) 2. A jump discontinuity at \(x = -1\) with the left limit of 0 and the right limit of 3 3. A point at \((1, 1)\) with a horizontal tangent for smooth connection coming from the left 4. A jump discontinuity at \(x = 1\) with a limit from the right of 4 The graph should now satisfy all the given properties.

Key Concepts

Graph Sketching

Graph sketching is an essential skill in understanding how a function behaves visually. It involves drawing a rough graph based on certain key properties without needing an exact formula. When sketching a graph, you should focus on:

• Marking important points where the function has specific values.
• Drawing lines or curves to show how the function approaches these points from different directions.
• Identifying and clearly showing any discontinuities, such as jumps or breaks in the graph.

In the given exercise, the graph of the function needs to reflect specific points and limits at certain values of \(x\). This means determining how the graph behaves as \(x\) approaches those values from the left (negative direction) and right (positive direction). Sketching involves carefully plotting these points and drawing curves that meet the conditions described by the limits and values.

Left-Hand Limit

The left-hand limit of a function at a particular point describes the value that the function approaches as \(x\) gets close to that point from the left side. It is a crucial concept for understanding the behavior of functions near points of interest, especially when sketching graphs.
In mathematical terms, the left-hand limit is expressed as \(\lim_{x \to a^-} h(x)\), which reads 'the limit of \(h(x)\) as \(x\) approaches \(a\) from the left.'
In our exercise:

• As \(x\) approaches \(-1\) from the left, \(h(x)\) approaches 0.
• This would be represented by drawing a line or curve that gets closer to the point \((-1, 0)\), but not actually touching it.
• As \(x\) approaches 1 from the left, \(h(x)\) approaches 1, joining smoothly with the value of the function at \((1, 1)\).

These left-hand limits help to define where the graph should be positioned and how the curves should look before reaching specific points.

Right-Hand Limit

The right-hand limit of a function tells us what value the function approaches as \(x\) nears a specific point from the right side. It complements the left-hand limit and helps us spot discontinuities or jumps.
Mathematically, the right-hand limit is written as \(\lim_{x \to a^+} h(x)\), indicating the behavior of \(h(x)\) as \(x\) comes towards \(a\) from values greater than \(a\).
For our exercise:

• As \(x\) approaches \(-1\) from the right, \(h(x)\) approaches 3. This requires a curve coming from the right that gets closer to \((-1, 3)\) but never actually reaches this point during the approach.
• When \(x\) nears 1 from the right, \(h(x)\) heads towards 4. So, a curve should be drawn approaching \((1, 4)\) from the right without actually touching it.

These details about right-hand limits help us anticipate breaks or sudden changes in the graph, like the ones seen in jump discontinuities.

Jump Discontinuity

A jump discontinuity occurs when there is a sudden 'jump' or 'gap' in the graph of a function. The left-hand and right-hand limits do not match at these points. The function value can suddenly 'jump' from one level to another, which creates a clear visible break on the graph.
During graph sketching, it's key to mark these discontinuities clearly:

• At \(x = -1\), the left-hand limit of 0 and right-hand limit of 3 indicate a jump. The function value jumps from \(h(x) = 0\) (left side) to 3 (right side) as \(x\) passes through \(-1\).
• At \(x = 1\), the left-hand limit is 1, matching the function value, but after \(x\) moves just past 1, \(h(x)\) jumps to approach 4, creating another jump discontinuity.

Jump discontinuities are critical features on a graph. They help illustrate the behavior of functions that defy smooth, continuous curves, requiring careful plotting to accurately represent these unique characteristics.