Question

In Exercises 31 and \(32,\) explain why the given function is continuous. $$f(x)=\frac{1}{x-3}$$

Short answer

The function \(f(x) = \frac{1}{x-3}\) is continuous for all real numbers except at \(x=3\) where the function is undefined.

Step-by-step solution

Identifying the function

The function provided is \(f(x) = \frac{1}{x-3}\).

Analyze the denominator

In order to find causes of discontinuity, you should analyze the denominator. A fraction is undefined where the denominator equals zero. So, the function \(\frac{1}{x-3}\) is undefined when \(x-3 = 0\). Solving this equation leads to \(x=3\). Therefore, the function is undefined at \(x=3\).

Determine the continuity

For all other values of \(x\), the function \(f(x) = \frac{1}{x-3}\) is continuous. Therefore, \(f(x)\) is continuous on its domain, which is all real numbers except \(x=3\). Hence, the function is not continuous at \(x=3\) but is continuous at all other real numbers.

Key concepts

Discontinuity

Imagine taking a peaceful walk along a path, and suddenly there’s a gap in the ground that you cannot cross — that’s similar to encountering a discontinuity in a function. In mathematics, a function is said to have a discontinuity when there is a point at which it is not continuous. A critical aspect of discontinuities is that they disrupt the flow of a graph. There are different types of discontinuities, but the one most relevant to our function, \( f(x) = \frac{1}{x-3} \), is called a point discontinuity, also known as a removable discontinuity.

This specific discontinuity occurs at a particular point where the function is not defined, which in this case is at \( x=3 \). At this point, you can think of the graph of \( f(x) \) as having a hole — it's like following the curve with a pencil, and suddenly you have to lift the pencil off the page because the function doesn't exist at that point. As you approach \( x=3 \) from either side, the values of \( f(x) \) get arbitrarily large, which visually would look like the graph is reaching upwards and downwards infinitely. Therefore, while \( f(x) \) is perfectly well-behaved at other points, at \( x=3 \) there is a break or jump, indicating a discontinuity.

Undefined Function

When speaking of an undefined function, we are referring to instances where the function does not have an output or value. In essence, it's like asking a question that has no answer. For the function \( f(x) = \frac{1}{x-3} \), the undefined behavior occurs where the denominator of the fraction, \( x-3 \), is equal to zero. When \( x \) equals 3, the function attempts to divide by zero, which is a mathematical operation that is undefined.

Mathematically, dividing by zero is off-limits because it can lead to contradictory results and does not align with the foundational properties of arithmetic. To visualize why a division by zero is a problem, imagine you have zero pieces of a pie; dividing these zero pieces among any number of people cannot be done — there’s nothing to divide! Similarly, the function cannot assign a real number as a value at this point, thereby making \( f(x) \) undefined at \( x=3 \). This undefined point leads to a type of discontinuity in the function as well.

Domain of a Function

The domain of a function is the set of all possible input values (usually \( x \)-values) that the function can accept without resulting in any undefined or non-real numbers. It’s like having a list of allowed questions for which the function can provide answers. For the given function \( f(x) = \frac{1}{x-3} \), we determine the domain by looking for values of \( x \) that do not cause any complications, such as division by zero.

In this case, the only problematic value is \( x=3 \), as it makes the denominator zero. Therefore, the domain of \( f(x) \) is all real numbers, except for \( x=3 \). We write the domain in interval notation as \( (-\infty, 3) \cup (3, +\infty) \), meaning that \( f(x) \) can take any real number except for 3. Understanding the domain is crucial for correctly graphing the function and for comprehending the behavior of the function across different intervals of the \( x \)-axis.