Question

Finding Discontinuities In Exercises \(73-76\) , use a graphing utility to graph the function. Use the graph to determine any \(x\) -values at which the function is not continuous. $$ f(x)=\left\\{\begin{array}{ll}{\frac{\cos x-1}{x},} & {x<0} \\ {5 x,} & {x \geq 0}\end{array}\right. $$

Short answer

The function \(f(x)\) has a discontinuity at \(x = 0\).

Step-by-step solution

Analyze the first part of the function for x

We first look at the part of the function defined for \(x < 0\), which is \(f(x) = \frac{\cos(x) - 1}{x}\). For \(x < 0\), we only need to consider the values of x where the denominator equals zero because this makes the function undefined. Since \(x\) doesn't equal zero in this range, the function is continuous for this part.

Analyze the second part of the function for x>=0

Next, analyze the part of the function defined for \(x \geq 0\), which is \(f(x) = 5x\). The linear function is continuous for all values of \(x\). Hence, there's no discontinuity in this part.

Checking the point of exchange

Lastly, check at \(x = 0\), the point where the function changes from one part to another. Plug \(x = 0\) into both parts of the function. For \(f(x) = \frac{\cos(x) - 1}{x}\), we get an undefined value (since we cannot divide by zero), but for \(f(x) = 5x\), we get \(0\). Hence, a discontinuity exists at \(x = 0\) as both parts of the function don't grant the same result.

Key concepts

Discontinuities in Piecewise Functions

Understanding discontinuities in piecewise functions is central to mastering calculus. Piecewise functions are functions defined by different expressions depending on the interval of the input value. Discontinuities occur when there is a 'break' in the function, which can be caused by a gap, jump, or a point where the function is not defined.

For example, examining the piecewise function given in the exercise, we have one function for values of \(x\) less than zero and another for \(x\) greater than or equal to zero. To find discontinuities, we should look for points where the function’s equations change, or where there might be gaps or holes. In particular, we're interested in any points where the function doesn't line up smoothly. In this case, the crucial point to examine is \(x = 0\), where the function switches from \(\frac{\cos x - 1}{x}\) to \(5x\).

When you graph the function or investigate the formulas, you need to verify if there's a smooth transition from one part to the next. If there is not – as in our example at \(x = 0\) – we can confirm the presence of a discontinuity. Tools like a graphing utility are particularly useful for visually identifying these points where the function behaves irregularly.

Graphing Utility in Calculus

A graphing utility in calculus is an invaluable tool for visualizing concepts and solving problems. It allows students to easily identify characteristics of functions, such as continuity, maxima, minima, and, importantly, discontinuities. By inputting a function into a graphing utility, one can quickly see where the function is not smooth or continuous.

Interpreting Discontinuities with Graphs

When using a graphing utility, a discontinuity may appear as a gap in the graph, a vertical asymptote, or a sudden jump from one value to another. It’s important to look closely at the graph around the expected points of discontinuity — often where the function’s formula changes.

By comparing the graphed function to its mathematical definition, one can substantiate the visual cues with analytical understanding. It’s not enough to just notice a disruption in the graph; one must analyze the function’s behavior around that point to distinguish a removable discontinuity from a jump or infinite discontinuity.

Continuity of Linear Functions

Linear functions are the simplest type of function when discussing continuity. These functions, which have the form \(f(x) = mx + b\), are continuous for all real numbers. They graph as straight lines with no gaps, jumps, or sudden changes in direction.

For an easy example, take the second part of the piecewise function in the exercise, \(f(x) = 5x\). This linear function is continuous because, for every value of \(x\), there’s a corresponding \(y\)-value, and as \(x\) increases or decreases, \(y\) does so in a consistent, predictable manner.

In contrast, non-linear parts of piecewise functions, especially those involving fractions, radicals, or logarithms, require more scrutiny since they have potential breaking points where continuity could be violated. Thus, recognizing that linear functions inherently carry no such risks simplifies the process of finding discontinuities in these contexts.