Question

Making a Function Continuous In Exercises \(61-66,\) find the constant \(a,\) or the constants \(a\) and \(b\) , such the function is continuous on the entire real number line. $$ f(x)=\left\\{\begin{array}{ll}{3 x^{3},} & {x \leq 1} \\ {a x+5,} & {x>1}\end{array}\right. $$

Short answer

The correct value for the constant \(a\) which makes the given piecewise function continuous over the entire real line is \(a = -2\).

Step-by-step solution

Calculating the Limit from Left Side

First, let's find the left limit as \(x\) approaches 1. This requires evaluating the function \(f(x) = 3x^3\) at \(x = 1\). Plug \(x = 1\) into the equation to find the left limit. So, \(f(1) = 3*(1)^3 = 3\).

Equating the Right to Left Limit

To make the function continuous at \(x = 1\), the function value at \(x =1\) from the left should be equal to the function value from the right. So, equate the right side of the function \(f(x) = ax + 5\) at \(x = 1\) to the left limit calculated in the previous step which is 3. That gives us: \(a*1 + 5 = 3\). Now you need to solve this equation for the unknown \(a\) to find the value of the constant \(a\) which will make the function continuous at \(x =1\).

Determining Constant \(a\)

Now, solve the equation \(a*1 + 5 = 3\) from the previous step for \(a\) . That gives \(a = 3 - 5 = -2\).

Key concepts

Limits

Understanding limits is crucial when analyzing how a function behaves as it gets close to a specific point. A limit helps us determine the value a function approaches as the input (usually denoted as \(x\)) approaches a certain number.
Knowing the limit is vital in ensuring a function is smooth and without breaks near that number. In our given exercise, to make the function continuous at \(x = 1\), we must first understand the left and right limits at this point.

• **Left-limit:** This is the value the function approaches when \(x\) comes from values less than \(1\). Here, we use the expression \(3x^3\), so we calculate \(3(1)^3 = 3\).
• **Right-limit:** This represents the value the function approaches when \(x\) comes from values more than \(1\). We find this using the piece-wise function \(ax + 5\). To ensure continuity, both the left and right limits should be equal.

By equating these, we ensure that the function does not "jump" at \(x = 1\). This is why finding limits on both sides is fundamental.

Piecewise Functions

Piecewise functions are fascinating because they are defined by different expressions for different intervals of the domain. This means the function can "piece" together different formulas depending on the input \(x\).
In the exercise, we deal with a piecewise function given by:

• \(f(x) = 3x^3\) for \(x \leq 1\)
• \(f(x) = ax + 5\) for \(x > 1\)

The challenge with piecewise functions is to make the entire function act smoothly at the point where the pieces meet, which in this case is \(x = 1\).
To tackle this challenge, both function pieces need to meet seamlessly at this point. That means the two equations must provide the same value when \(x = 1\). Solving for the values of \(a\) or potentially other constants in the piecewise parts helps ensure smooth transitions, avoiding any abrupt changes in the function's path.

Continuity at a Point

Continuity at a point occurs when a function is smooth and unbroken at that specific point. For a function to be continuous at \(x = c\), three conditions must be met:

• The function \(f(x)\) is defined at \(x = c\).
• The limit of \(f(x)\) as \(x\) approaches \(c\) exists.
• The limit equals the function value at \(x = c\), meaning \(\lim_{x \to c} f(x) = f(c)\).

In the exercise, to ensure continuity at \(x = 1\), we evaluated both the left and right limits using the piecewise definition. By solving: \[ ax + 5 = 3 \] at \(x = 1\), we ensured the function is smooth and without gaps at this critical point.
After solving, we found that \(a = -2\), ensuring both the left-hand and right-hand limits match the function value at \(x =1\), confirming continuity. Ensuring a function is continuous at a point not only maintains mathematical rigor but also ensures the function has real-world applications where smooth data or processes are modeled.