Question

Group Activity In Exercises \(37-40\) , verify that the function is continuous and state its domain. Indicate which theorems you are using, and which functions you are assuming to be continuous. $$y=\left|x^{2}-4 x\right|$$

Short answer

The function \(y=|x^{2}-4x|\) is continuous everywhere in its domain. The domain of the function is all real numbers.

Step-by-step solution

Identify the function

The function that we have is \(y=|x^{2}-4x|\). This is an absolute value function that contains a quadratic function \(x^{2}-4x\) inside.

Check for continuity

To check if the function is continuous, we need to confirm that it does not have any point in its domain where the left-hand limit, right-hand limit, and the function value are not equal. However, as we are dealing with an absolute value function that contains a polynomial function inside, we know that this type of function is always continuous at every point in its domain. Hence, the function \(y=|x^{2}-4x|\) is continuous everywhere within its domain.

Determine the domain

In this step, we're finding the set of all possible inputs for the function. For the given function, the domain is all real numbers, because the expression inside the absolute value function, \(x^{2}-4x\), can take on any real number value.

State the theorems used

The main theorem used here is the theorem that states that the absolute value function is continuous for every point in its domain. We also used the concept of domain of polynomial functions, which states that polynomial functions are defined for all real numbers.

Key concepts

Domain of a Function

Understanding the domain of a function is crucial when studying continuity and other aspects of functions. It represents the set of all possible input values (or 'x' values) for which the function is defined.

In simpler terms, the domain is like the guest list for a party; it's a list of everyone who is 'allowed' to enter. For most polynomial functions, like the quadratic function within the absolute value in our exercise \( x^2 - 4x \), the domain is all real numbers, denoted as \( \mathbb{R} \). This is because you can plug in any real number into a polynomial and get out a valid result.

How Do We Determine the Domain?

For a function to have a particular number in its domain, the function must be able to produce a real number when we plug that value into the function. When complexities such as division by zero or square roots of negative numbers are involved, the domain might be restricted.

In the function \( y = |x^2 - 4x| \) from our exercise, since there's no division or square roots, we can determine quickly that its domain includes all real numbers. No matter what value of 'x' you choose, the function can handle it, leading us to this broad domain.

Continuity of Polynomial Functions

To address the continuity of polynomial functions, we should first grasp what makes a function continuous. Imagine drawing the function on a piece of paper without lifting your pen; if you can do this throughout the function's domain, it is said to be continuous.

Polynomial functions, like the one inside our exercise's absolute value \( x^2 - 4x \), are the epitome of this kind of continuity because they do not have breaks, jumps, or holes in their graphs. This can be attributed to the smooth, predictable nature of their equations.

Why Are They Continuous Everywhere?

Since polynomial functions are composed of terms like \( x^n \), where 'n' is a non-negative integer, and constants, combined through addition, subtraction, and multiplication, they naturally produce smooth curves. These operations don't introduce any abrupt changes that could break the graph, ensuring their continuousness across the entire set of real numbers, their domain. As a result, within its domain, our function \( y = |x^2 - 4x| \) inherits this continuity seamlessly.

Limits and Continuity

The concepts of limits and continuity are closely intertwined in calculus. Both help us understand the behavior of functions at specific points and over intervals.

What Are Limits?

Limits describe what happens to the value of a function as its input (or 'x' value) gets closer to a certain point. Essentially, they tell us the function's destination, even if we never arrive exactly at that point.

For instance, as 'x' approaches a particular value, if the function gets arbitrarily close to a particular 'y' value, then the function has a limit at that 'x' value. On the other hand, continuity takes this concept a step further.

Understanding Continuity

A function is continuous at a point if the limit exists at that point and is equal to the function's value there. That's like saying, as you walk towards a doorway (the limit), you can actually step through it into the room (the function's value) without any obstructions. In our exercise, \( y = |x^2 - 4x| \) is continuous because at any point within its domain, the left-hand and right-hand limits exist and are equal to the function's value at that point. This absence of obstructions in its graph fortifies our understanding of continuity for absolute value functions, especially when built upon polynomial expressions.