Question

Sketching functions a. Sketch the graph of a function that is not continuous at \(1,\) but is defined at 1 b. Sketch the graph of a function that is not continuous at \(1,\) but has a limit at 1

Short answer

Question: Sketch two different types of functions: one with a removable discontinuity at x=1 and one with a limit at x=1. Describe the graphs of the functions in the context of the continuity and discontinuity. Answer: The removable discontinuity can be represented by the function f(x) = {x if x≠1, 2 if x=1}. The graph of this function will be a continuous straight line y=x with a gap at the point (1,2). The limit point discontinuity can be represented by the function g(x) = (x^2-1)/(x-1). The graph of this function will be a continuous straight line y=x+1 with an open circle at the point (1,2) representing the discontinuity at x=1.

Step-by-step solution

Sketching Function with a Removable Discontinuity

Let's consider a simple example of a function that has a removable discontinuity at \(x=1\), for instance: $$f(x)=\begin{cases} x, & \text{if } x \neq 1\\ 2, & \text{if } x=1 \end{cases}$$ To sketch the graph of the function, we will follow these steps: 1. Draw the line \(y=x\) for all \(x\) values except \(x=1\). 2. The function value at \(x=1\) is \(2\). Mark this point on the graph. 3. Join the line segments ensuring that the point \((1,2)\) is not connected to the line \(y=x\). This represents the discontinuity. The resulting graph should have a small gap at the discontinuity \((1,2)\) on an otherwise continuous line \(y=x\).

Sketching Function with a Limit Point

Now let's consider a function that has a limit point at \(x=1\) but is not continuous at that point. A simple example would be: $$g(x)=\frac{x^2-1}{x-1}$$ To sketch the graph of the function, we will follow these steps: 1. Simplify the function to find its limit as \(x\) approaches \(1\): $$\lim_{x\to1} \frac{x^2-1}{x-1} = \lim_{x\to1} \frac{(x-1)(x+1)}{x-1} = \lim_{x\to1} x+1 = 2$$ 2. Since the function is undefined at \(x=1\), we will need to remove this point from the graph. 3. Draw the line \(y=x+1\) for all \(x\) values except \(x=1\). 4. Highlight the limit point by drawing an open circle at \((1,2)\). The resulting graph should have an open circle at the point \((1,2)\) on an otherwise continuous line \(y=x+1\).

Key concepts

Removable Discontinuity

A removable discontinuity occurs in a function where a point is "missing" on the graph, but the graph would be continuous if this point were included. This usually happens because a function is defined at a point in such a way that it's disconnected from the rest of the graph. For example, consider the function given in the solution:\[ f(x) = \begin{cases} x, & \text{if } x eq 1 \ 2, & \text{if } x = 1 \end{cases} \]In this function, other than at \(x=1\), the graph follows the line \(y=x\). However, at \(x=1\), the value jumps to \(y=2\), creating a small gap or a point that "breaks off" the continuous line. This break is the removable discontinuity. To represent it graphically, we sketch the line \(y=x\) but leave a gap at \((1, 1)\) and instead mark the point \((1, 2)\). The discontinuity is 'removable' because if we'd redefine \(f(x)\) such that \(f(1) = 1\), the graph would then be continuous.

Limit of a Function

The limit of a function describes the behavior of the function as it approaches a particular point. This is important for understanding how the function behaves near discontinuities. We often use this concept to explore what a function "tends" towards as its input values approach a specified point.For instance, let's consider the function:\[ g(x) = \frac{x^2-1}{x-1} \]The function appears to be undefined at \(x = 1\) because it leads to a division by zero. However, we can simplify the expression:\[\lim_{x \to 1} \frac{(x-1)(x+1)}{x-1} = \lim_{x \to 1} (x+1) = 2\]This simplification shows that as \(x\) gets very close to 1 from either side, the value of \(g(x)\) approaches 2. Thus, the limit of \(g(x)\) as \(x\) approaches 1 is 2. Even though the function is not defined at this point, this concept allows us to predict the behavior of the function around \(x=1\), helping us understand its graphical representation.

Graph Sketching

Graph sketching involves plotting a function on a coordinate grid to visually depict its behavior, including any discontinuities or limits. To effectively sketch a graph, follow these steps:

• First, consider the defined values of the function. Identify where it is continuous and where it has any form of discontinuity, whether removable or otherwise.
• Pay close attention to points of discontinuity. Use open circles to indicate points that belong to the graph but are not defined within the function, and solid dots for clearly defined points.
• The line or curve created by the function must be drawn excluding any points of discontinuity to ensure they are clear, especially in cases with a removable discontinuity.

In the example of \(g(x) = \frac{x^2-1}{x-1}\), after simplification and understanding its limit, the graph resembles the line \(y=x+1\) with an open circle at \((1, 2)\). For the function \(f(x)\) with a removable discontinuity, the graph is the line \(y=x\) with a gap at \(x=1\) and a solid dot at \((1, 2)\). Graph sketching provides a powerful way to visualize functions, highlighting their characteristics and behavior.