Question

In Exercises \(11-18,\) use the function \(f\) defined and graphed below to answer the questions. $$f(x)=\left\\{\begin{array}{ll}{x^{2}-1,} & {-1 \leq x<0} \\\ {2 x,} & {0 < x < 1} \\ {1,} & {x=1} \\ {-2 x+4,} & {1 < x < 2} \\ {0,} & {2 < x < 3}\end{array}\right.$$ (a) Is \(f\) defined at \(x=2 ?\) (Look at the definition of \(f . )\) (b) Is \(f\) continuous at \(x=2 ?\)

Short answer

The function \(f\) is not defined at \(x = 2\), and hence is not continuous at \(x = 2\).

Step-by-step solution

Determine if \(f\) is defined at \(x=2\)

Looking at the definition of \(f(x)\), we find that the function is defined as \(0\) when \(2 < x < 3\). However, the function is not defined exactly at \(x=2\). Hence, \(f\) is not defined at \(x = 2\).

Determine if \(f\) is continuous at \(x=2\)

To be continuous at a point, the function must first be defined at that point. Since \(f\) is not defined at \(x = 2\), we cannot speak of continuity at this point.

Key concepts

Defining Functions

In mathematics, the concept of defining functions is fundamental to understanding how to express relationships between variables.
A function can be thought of as a machine that gives out an output for every input you put into it. But for this machine to work correctly, we must establish clear instructions for every possible input. This is what defining a function means: providing a rule or formula that specifies how to compute the value of the function for each input within a certain domain.

Piecewise Functions

In the case of piecewise functions, the 'instructions' can change depending on the interval the input falls into. The term 'piecewise' indicates that different 'pieces' of the function have their own defining expressions. For example, in the given exercise, the function f(x) is defined by different algebraic expressions for different intervals of x. Properly defining a piecewise function means stating these intervals and associated expressions explicitly. It's also crucial to include points where the function changes its rule, ensuring that every input in the domain has a well-defined output.

Continuity of Functions

Understanding the continuity of functions is key to grasping many concepts in calculus and mathematical analysis.
A function is said to be continuous at a point if there is no interruption in its graph at that point. Specifically, a function f(x) is continuous at a point x = c if the following three conditions are met:

• The function is defined at x = c.
• The limit of f(x) as x approaches c exists.
• The limit of f(x) as x approaches c is equal to f(c).

Intuitive Understanding of Continuity

Think of drawing the graph of a function without lifting your pencil off the paper. If you can do this for the entire range of your function, then your function is continuous in that interval. In contrast, if you encounter a point where you must lift your pencil to continue drawing, this suggests a lack of continuity at that point. Applying this concept to the textbook example, we observe that because the function is not defined at x=2, we would have to lift our pencil off the page, indicating a break in continuity.

Limits and Continuity

Limits play a pivotal role in the study of calculus, serving as the foundation for defining continuity, derivatives, and integrals.
The limit of a function as x approaches a specific value is what the function value 'approaches' or gets close to as the input x gets close to that value. It doesn't have to be what the function actually equals at that point—if the function is even defined there—but rather where it is 'heading'.

Importance of Limits in Continuity

Limits help determine if a function 'behaves nicely' as inputs approach a particular value. For continuity at a point x = c, we require that the function not only be defined at c but also that the limit of the function as x approaches c from both the left and the right exists and equals the function's value at c, denoted as f(c). Applying this principle to the exercise given, since the function f(x) isn't defined at x=2, there is no limit to consider at this specific point, and thus it is not continuous there.