Question

In Exercises \(51 - 54 ,\) complete parts \(( a ) , (\) b) \(,\) and \(( c )\) for the piecewise-defined function. $$c = - 1 , f ( x ) = \left\\{ \begin{array} { l l } { 1 - x ^ { 2 } , } & { x \neq - 1 } \\ { 2 , } & { x = - 1 } \end{array} \right.$$

Short answer

a) The value of the function at \(c = -1\) is \(2\). b) The function is not continuous at \(c = -1\). c) The graph of the function includes the curve \(1 - x^{2}\) for all \(x \neq -1\) and a point at \((-1,2)\).

Step-by-step solution

Find the value of the function at a specific point

Firstly, to solve part \(a\), let's determine the value of \(f(c)\) when \(c = -1\). Looking at the piecewise function definition, it is seen that if \(c = -1\), then \(f(c) = 2\). So the value of the function at \(c = -1\) is \(2\).

Determine continuity

For part \(b\), let's determine whether the function is continuous at \(c = -1\). A function is continuous at a point if the function is defined at that point and the limit from the left and the right are both equal to the function's value at that point. First, find the limit as \(x\) approaches \(-1\) from the left (\(x \neq - 1\)). Using the given function \(f(x) = 1 - x ^ { 2 }\) the result is \(1 - (-1) ^ { 2 } = 1 - 1 = 0\). The limit from the right is not so straightforward because there's no function defined for values of \(x\) greater than \(-1\). Therefore, we can reason that the limit as \(x\) approaches \(-1\) from the right doesn't exist. As folllow, the function is not continuous at \(c = -1\) because the limits from different directions are not equal.

Graph the function

In part \(c\), let's graph the function. On the x-y plane, draw the curve of \(f(x) = 1 - x^{2}\) for all \(x\) except \(-1\). Then clearly indicate that at \(x = -1\), the value of \(f(x)\) is \(2\), which can be indicated with a filled circle at the point \((-1,2)\). Notice that at \(-1\), the function has a gap (it jumps from the curve to the point), indicating that it is indeed not continuous at that point.

Key concepts

Continuity of Functions

The concept of continuity in functions is crucial for understanding how they behave at specific points or intervals. A function is said to be continuous at a point if, intuitively speaking, you can draw the function at this point without lifting your pencil. Mathematically, a function is continuous at point c if the following three conditions are met:

• The function is defined at c.
• The limit of the function as x approaches c exists.
• The limit of the function as x approaches c is equal to the function's value at c.

In the given piecewise-defined function example, we can see that although the function is defined at c = -1 and has a value of f(c) = 2, the limit from the left as x approaches -1 is 0, and there is no function defined for values from the right, which means the limit from the right doesn't exist. Therefore, the function is not continuous at c = -1 because the limits from different directions are not equal, violating the conditions for continuity.

Limits of Functions

Exploring the limits of functions helps us understand the behavior of functions as they approach a particular value. It is the value that a function f(x) tends to as the variable x approaches a specific number. The limit does not need the function to be defined at that point.

For the piecewise function in the exercise, the limit as x approaches -1 from the left is 0 because substituting x in the function f(x) = 1 - x^2, we get f(-1) = 0. However, as there is no piece of the function defined to the right of x = -1, we don't have a limit from the right. When the limits from both sides of a point differ or if one of them doesn’t exist, the overall limit at that point doesn't exist. Thus, you can conclude that the limit of f(x) as x approaches -1 does not exist due to different-sided behavior around the point.

Graphing Functions

Graphing provides a visual representation that helps to understand the behavior and properties of a function. To graph a piecewise-defined function like the one presented in the exercise, you plot different parts of the function based on the conditions defined for different intervals.

In this case, the function f(x) = 1 - x^2 would be graphed normally, except at x = -1. When graphing, it's essential to denote that at x = -1, the function does not follow the rule 1 - x^2 but instead takes the value of 2. This can be shown on the graph as a 'hole' or an open circle at x = -1 on the curve y = 1 - x^2, and a separate filled circle at the point (-1, 2) to represent the value of the function at x = -1. By graphing the function this way, it becomes evident where the function is not continuous.