Question

In Exercises 27 and 28, sketch a graph of a function that satisfies the given values. (There are many correct answers.) $$ \begin{array}{l}{f(0) \text { is undefined. }} \\ {\lim _{x \rightarrow 0} f(x)=4} \\ {f(2)=6} \\ {\lim _{x \rightarrow 2} f(x)=3}\end{array} $$

Short answer

The graph of the function has: a vertical asymptote at \(x=0\) which approaches the \(y=4\) from both sides, a point at \(x = 2\) with value \(f(2) = 6\), and a open circle at the point \(2,3\) where the function's value gets close as \(x\) approaches 2, but skips to 6 at that exact point.

Step-by-step solution

Understanding the Problem

For this problem, we must plot a function that satisfies the given values. To understand, let's break down what each condition means: \n - \(f(0)\) is undefined: This means that the function \(f(x)\) does not have a value at \(x = 0\). This is often indicated on a graph by an open circle or vertical asymptote.\n- \(\lim_{x \rightarrow 0} f(x)=4\) : This indicates that as \(x\) approaches 0, the function's value gets closer to 4. This means that while the function itself isn't defined at 0, any value we choose that is very close to but not equal to 0, would give us something very close to 4.\n- \(f(2)=6\) : This simply means that at \(x = 2\), the function's value is 6.\n- \(\lim_{x \rightarrow 2} f(x)=3\) : This indicates that as \(x\) approaches 2, the function's value gets closer to 3. However, unlike the first lim function, this function is defined at \(x = 2\) and has a different value. Thus, we need to indicate this in our graph as well.

Sketching the Graph

First, put a vertical asymptote at \(x=0\) to represent that \(f(0)\) is undefined. Then, draw a horizontal asymptote at \(y=4\) that approaches from both sides to represent the limit as \(x\) approaches 0. The function does get very close to 4, but never quite gets there since it isn't defined at 0.\nSecond, put a point at \(f(2) = 6\), representing that the function's value is 6 when \(x = 2\).\nFinally, demonstrate that the limit of \(f(x)\) is 3 as \(x\) approaches 2. This is done by drawing lines approaching the point (2,3) from both the left and right direction. However, indicate with an open circle that although the function gets close to 3, it jumps to 6 at the exact point \(x = 2\).

Key concepts

graph sketching

Graph sketching is a valuable skill in mathematics that involves creating a visual representation of a function based on its characteristics and given conditions. When sketching graphs, you are not necessarily producing an exact plot, but more of a rough illustration showing the general behavior of the function.
For instance, when given specific values and limits, the task is to plot the function in a way that corresponds with this information. With a condition like \( f(0) \text{ is undefined} \), you know there's either a gap or a special feature like an asymptote at \( x = 0 \). For \( \lim_{x \to 0} f(x) = 4 \), the value of the function approaches 4 as \( x \) nears 0, guiding how you sketch the curve close to this point. Always remember that these sketches provide a simplistic, intuitive overview rather than precise detail.
Mastering graph sketching involves practicing how to depict these conditions with lines and points, highlighting key transitions and behavior changes of the function. It helps to ensure readability and clarity, using open circles and asymptotic lines where necessary.

vertical asymptote

A vertical asymptote is a line where the graph of a function tends to infinity as it approaches a certain \( x \) value, rather than reaching a finite value. Vertical asymptotes arise in functions where there is division by zero as \( x \) approaches the asymptotic value.
In the context of our exercise, \( f(0) \text{ is undefined} \) suggests a vertical asymptote at \( x = 0 \). However, it is essential to identify that because the limit, \( \lim_{x \to 0} f(x) = 4 \), is finite, the function approaches the same value from both sides of the asymptote. This illustrates to us that despite the function not being defined at \( x = 0 \), it closely approaches a common value, making the behavior of the function smooth and predictable.

• Vertical asymptotes are depicted using dashed vertical lines on graphs.
• They represent places where the function doesn't touch or cross.

Understanding vertical asymptotes helps you better grasp the long-term behavior and restrictions of a function as it relates to specific \( x \) values.

horizontal asymptote

Horizontal asymptotes are lines that the output value of a function approaches as the input approaches infinity or a specific point. Unlike vertical asymptotes, which pertain to \( x \)-values, horizontal asymptotes usually focus on \( y \)-values.
In our scenario, a horizontal asymptote helps illustrate how \( f(x) \) approaches a certain \( y \)-value without necessarily touching it. Specifically, the limit \( \lim_{x \to 0} f(x) = 4 \) indicates that while \( f(x) \) is not defined at \( x = 0 \), from both sides, it nears a \( y \)-value of 4, showcasing smooth adjacency without contact.
Horizontal asymptotes act as an insight into the function's end behavior and help us understand how a function behaves as it moves toward specific x-values:

• They are often revealed by limits as \( x \to \text{some value} \)
• Horizontal lines represent them on graphs, suggesting approach not contact.

Grasping horizontal asymptotes lets you better predict functional behavior at different sections, particularly how it behaves far from the central points of interest.

undefined function value

An undefined function value signifies a point in the function where it does not have a specific output. In mathematical terms, this means that for a certain \( x \)-value, \( f(x) \) cannot be calculated or doesn't exist.
For our task, \( f(0) \text{ is undefined} \) immediately tells you that something unique occurs at \( x = 0 \), potentially impacting the function's behavior there. This is important because when limits like \( \lim_{x \to 0} f(x) = 4 \) exist, they show that although the function doesn't define at that \( x \)-value, it approaches a specific \( y \)-value very closely when nearing that point.

• Undefined values hint at breaks or jumps in the graph.
• Often results from operations like division by zero or other limits.

Understanding undefined values ensures that you note key disrupts or skips in function plots while considering continuity and overall graph appearance. This knowledge lets you intuit why a function might break or change radically in its plot.