Question

Find the constants \(a\) and \(b\) such that the function is continuous on the entire real number line. $$ f(x)=\left\\{\begin{array}{ll} x^{3}, & x \leq 2 \\ a x^{2}, & x>2 \end{array}\right. $$

Short answer

The constant \(a\) is 2.

Step-by-step solution

Find the left-hand limit

The left-hand limit is the function's value when \(x = 2\). For the given function, this is \(2^3\) or 8.

Find the right-hand limit

The right-hand limit is the function's value when \(x = 2\). For the given function, this is \(a \cdot 2^2\) or \(4a\).

Set left-hand limit equal to right-hand limit

Since a continuous function's left-hand limit at a given point equals its right-hand limit at that point, we set 8 equal to \(4a\) and solve for \(a\). This gives us \(a = 2\).

Key concepts

Limits of Functions

Understanding the concept of limits is foundational in calculus. The limit of a function describes the behavior of a function as it approaches a specific point from both directions. This is important because not all functions will have a defined value at a certain point, but we can talk about what value the function is getting close to as we approach that point.

Consider the expression \(\lim_{x \to c} f(x)\), which represents the limit of the function \(f(x)\) as \(x\) approaches \(c\). In the case of a continuous function, the limit exists and is equal to \(f(c)\). However, if a function approaches different values from the left and the right, the limit at that point does not exist, signaling a discontinuity.
For example, in the provided exercise, finding the limit of \(f(x)\) as \(x\) approaches 2 from both directions helps us determine the constant \(a\) that will make the function continuous. We calculated these as the left-hand limit (\(8\)) and the right-hand limit (\(4a\)). To achieve continuity, these two must be equal, leading us to find \(a=2\).

Piecewise Functions

Piecewise functions are functions defined by multiple sub-functions, each of which is applicable to a certain interval of the function's domain. Each 'piece' of the function is defined over a specific interval and might have a different functional form.

In the context of our exercise, \(f(x)\) is a piecewise function with two pieces. For values of \(x\) less than or equal to 2, \(f(x)\) is defined as \(x^3\), and for values greater than 2, it's defined as \(ax^2\). When transitioning from one piece to another, a piecewise function can either be continuous or discontinuous, depending on whether the output of the function at the boundary connects the two pieces seamlessly.

To ensure the function is continuous along its entire domain, the outputs from the pieces must be equal where they meet—at \(x=2\) in our case. This is why we find the limits at this junction and set them equal to identify \(a\), one of the constants needed for continuity.

Continuity in Calculus

Continuity is a hallmark of calculus, where a function \(f(x)\) is called continuous at a point \(x=c\) if the following conditions are met: firstly, \(f(c)\) is defined; secondly, the limit of \(f(x)\) as \(x\) approaches \(c\) exists; and thirdly, the limit of \(f(x)\) as \(x\) approaches \(c\) is equal to \(f(c)\). If these conditions hold for every point in a function's domain, then the function is continuous over its entire domain.

A function that is not continuous is said to have a discontinuity, which can be a jump, infinite, removable, or other types of discontinuity, depending on the nature of the break in continuity.
In solving our exercise, we looked for the constants \(a\) and \(b\) that would make a piecewise function continuous across the entire real number line. Through the limits we evaluated, we determined \(a=2\) to ensure these continuity conditions are satisfied, thus closing any potential gaps and creating a smooth transition between the pieces of the function.