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Chapter 11: Problem 47

If a function is not defined at \(a,\) how is this shown on the function's graph?

Short Answer

Expert verified
On a function's graph, a function not defined at a point 'a' is typically shown by a hole or an open circle.

Step by step solution

01

Understanding the concept

A function can be nonexistent or undefined at a certain point 'a' for several reasons. This could be because the function does not have a specific value at that point or it could also be due to a discontinuity, e.g., because of division by zero.
02

Graphical Representation

On a graph, an undefined point is usually represented by a hole or an open circle. This shows that the function does not exist at that point 'a'
03

Interpretation

In a graphical representation, when you see a hole or an open circle in the line or curve, you can interpret that as the point at which the function is undefined.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphical Representation
When we talk about the graphical representation of a function, we are referring to the visual depiction of the function's behavior on a coordinate plane. Graphs help us easily see where a function is defined or undefined.
An undefined function at a specific point is typically shown as a 'hole' or an 'open circle' on its graph.
  • Think of it like a missing puzzle piece; the overall picture (the graph) is disrupted at that point.
  • For example, if you were graphing a function that is undefined at a point 'a', you'd draw the curve normally except for an open circle where 'a' would appear on the x-axis.
Seeing this on the graph immediately tells you that the function does not exist exactly at that point.
This representation is crucial for understanding the actual domain of the function 鈥� what x-values are allowed, and which are not.
Discontinuity
Discontinuity is a concept that describes a sudden break or hole in a function's graph. Discontinuities occur at points where the function is undefined.
There are different types of discontinuities, but a common one is a removable discontinuity, often resulting from division by zero.
  • A removable discontinuity can be "fixed" by redefining the function over the problematic point, but it usually remains represented by a hole in the graph.
  • This hole signifies that, although the graph is continuous before and after this point, it is not defined at this single location.
Understanding where these discontinuities occur helps us better understand the behavior of the function and the continuity of its domain.
Typically, resolving discontinuities involves factoring, analyzing limits, or adjusting definitions of functions.
Division by Zero
Division by zero is one of the most common reasons why a function is undefined at a specific point. It occurs when the denominator of a fraction in the function equals zero, creating an undefined expression.
  • Mathematically, you cannot divide a number by zero, as it leads to a contradiction. This is what causes functions to be undefined at particular x-values.
  • In practical terms, if you come across an undefined point in a function due to division by zero, this is likely depicted by a gap on its graph.
Identifying such points is essential to understanding and working with functions, particularly when graphing or solving them.
Recognizing and acknowledging these undefined points ensure the correct interpretation and representation of the function's behavior.

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Most popular questions from this chapter

Use properties of limits and the following limits $$\begin{array}{lc}\lim _{x \rightarrow 0} \frac{\sin x}{x}=1, & \lim _{x \rightarrow 0} \frac{\cos x-1}{x}=0 \\\\\lim _{x \rightarrow 0} \sin x=0, & \lim _{x \rightarrow 0} \cos x=1\end{array}$$ to find the indicated limit. $$\lim _{x \rightarrow 0} \frac{\tan x}{x}$$

Verify the identity: $$ \sin \left(x+\frac{\pi}{6}\right)+\cos \left(x+\frac{\pi}{3}\right)=\cos x $$

Consider the function \(f(x)=3 x+2 .\) As \(x\) approaches 1 \(f(x)\) approaches \(5: \lim _{x \rightarrow 1} f(x)=5 .\) Find the values of \(x\) such that \(f(x)\) is within 0.1 of 5 by solving $$ |f(x)-5|<0.1 $$ Then find the values of \(x\) such that \(f(x)\) is within 0.01 of 5

Graph \(y=3 \sin 2 x .\) Then use the graph to obtain the graph of $y=3 \csc 2 x .

What does the limit notation \(\lim _{x \rightarrow a} f(x)=L\) mean?
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