Problem 67 - Calculus and its applications

Question

Is the function given by \(g(x)=\frac{1}{x+5}\) continuous over the interval (-4,4) ? Why or why not?

Step-by-Step Solution

Answer

The function is continuous on the interval (-4, 4) because it's defined and continuous at every point within this interval.

Step 1: Understand Continuity

A function is continuous on a given interval if, for every point in that interval, the function is defined and does not have any interruptions, jumps, or holes. Mathematically, a function is continuous at a point \(x = a\) if \(\lim_{x \to a} g(x) = g(a)\).

Step 2: Find Domain of the Function

The function \(g(x) = \frac{1}{x+5}\) is defined for all \(x\) except where the denominator is zero. Thus, set \(x+5=0\) to find the point of discontinuity. Solving this gives \(x = -5\). The function is undefined at \(x = -5\), so it's not continuous there.

Step 3: Check Interval (-4, 4)

The interval (-4, 4) does not include \(x = -5\). Hence, within this interval specifically, the function is defined for every point since the problematic point \(x = -5\) is outside the interval.

Step 4: Conclude on Continuity

As there are no points within the interval (-4, 4) where the function \(g(x) = \frac{1}{x+5}\) is undefined, the function is continuous on this interval. It follows that the function does not have any interruptions on (-4, 4).

Key Concepts

Domain of a Function

The domain of a function is a set of all possible input values, which allow the function to produce an output. For the function to work mathematically, we exclude values that make the function undefined.
In the case of the function \(g(x) = \frac{1}{x+5}\), it is crucial to ensure the denominator doesn’t become zero. When the denominator becomes zero, the function becomes undefined since division by zero is not possible.
To find these restricted values, set the denominator equal to zero and solve for \(x\). Therefore, \(x + 5 = 0\) gives \(x = -5\). Hence, the domain of \(g(x) = \frac{1}{x+5}\) is all real numbers except \(x = -5\).
In interval notation, this is expressed as \(( -\infty, -5 ) \cup ( -5, \infty )\).

Point of Discontinuity

A point of discontinuity is a value in the domain where a function is not continuous. This typically occurs where the function is undefined or there are jumps or breaks in its graph.
For the function \(g(x) = \frac{1}{x+5}\), we already identified \(x = -5\) as the number where the denominator equals zero. This results in the function being undefined at this point. Hence, \(x = -5\) is a point of discontinuity.
Because of this discontinuity, the function has an interruption, and its graph doesn’t appear smooth and connected at this point. However, since this occurs at \(x = -5\) and not within the interval (-4, 4), the function is continuous over the given interval.

Interval Notation

Interval notation is a way of representing a set of numbers along the number line using pairs of numbers to indicate where the interval begins and ends.
This notation is used to express continuous blocks of numbers, with different symbols capturing whether endpoints are included or not:

• Round brackets \(()\), indicate that endpoints are not included, known as an "open" interval.
• Square brackets \([]\), indicate that endpoints are included, known as a "closed" interval.

Let's consider the interval (-4, 4). The round brackets mean that -4 and 4 are not part of the interval itself. This choice effectively excludes any problematic points happening exactly at these boundary values.
In the context of the function \(g(x) = \frac{1}{x+5}\), using interval notation helps clearly communicate that the function is evaluated only where it's defined and continuous, providing a clear, universal way to express these properties.