Problem 70 - Calculus and its applications

Question

Is the function given by \(f(x)=\frac{1}{x}+3\) continuous over the interval (-7,7) ? Why or why not?

Step-by-Step Solution

Answer

The function is not continuous over (-7, 7) because it is undefined at x = 0.

Step 1: Understanding Continuity

A function is continuous over an interval if it is continuous at every point in the interval. For the function \(f(x) = \frac{1}{x} + 3\), we need to check its continuity across the interval \((-7, 7)\).

Step 2: Check for Discontinuities

The function \(f(x) = \frac{1}{x} + 3\) will be discontinuous where the denominator, \(x\), is zero since division by zero is undefined. Thus, we check if there is any point within the interval \((-7, 7)\) where \(x = 0\).

Step 3: Identifying Problematic Point

In the function \(f(x) = \frac{1}{x} + 3\), the problematic point is \(x = 0\). This point lies within the interval \((-7, 7)\), causing the function to be undefined at \(x = 0\).

Step 4: Conclusion About Continuity

Since the function is undefined and discontinuous at \(x = 0\), \(f(x) = \frac{1}{x} + 3\) is not continuous over the interval \((-7, 7)\) as continuity is broken at \(x = 0\).

Key Concepts

Discontinuities

In calculus, a discontinuity occurs in a function when there is an interruption in the graph, causing the function not to be defined or not to be smooth at a particular point. For the function \( f(x) = \frac{1}{x} + 3 \), discontinuities arise when the denominator of any fraction is equal to zero, leading to an undefined situation. In this case, the denominator is \( x \) and becomes zero at the point \( x = 0 \).
When we graph such functions, a gap or a hole is often seen where the discontinuity lies. Discontinuities can be of various types such as removable discontinuities, jump discontinuities, and infinite discontinuities. Understanding these types helps anticipate how a function behaves at or around certain points. For the given function \( f(x) \), the discontinuity is infinite at \( x = 0 \) as the value of the function would approach infinity near this point.

Intervals in Calculus

An interval in calculus is a range of input values (or \( x \)-values) where you study the behavior of a function. This range can be open, closed, or half-open, depending on whether you include the endpoints or not.
In this exercise, we are examining the interval \((-7, 7)\) for the function \( f(x) = \frac{1}{x} + 3 \). An open interval, indicated by parentheses, means that the endpoints \(-7\) and \(7\) are not included in the interval. Within this range, we check every single point to determine the function's continuity.
Because of the open interval, we only need to worry about the values strictly between \(-7\) and \(7\), not covering the endpoints. It's crucial to look for any breaks or undefined points across this interval which might affect continuity.

Undefined Functions

A function becomes undefined at points where its expression cannot produce a real number result. This often happens when you try to divide by zero or take an even root of a negative number.
In the function \( f(x) = \frac{1}{x} + 3 \), it is undefined specifically at \( x = 0 \), since dividing by zero is impossible. This lack of definition at a specific point within an interval means the function cannot be continuous across that interval.
In any calculus problem, identifying these undefined points is crucial as they directly cause discontinuities. By understanding where a function is undefined and its impact on intervals, you can better grasp the overall behavior and characteristics of mathematical functions.