Problem 26 - Thomas Calculus

Question

At what points are the functions in Exercises 13-30 continuous? $$y=\sqrt[4]{3 x-1}$$

Step-by-Step Solution

Answer

The function is continuous for \( x \geq \frac{1}{3} \).

Step 1: Identify the Function Type

The function given is a fourth root function, specifically \( y=\sqrt[4]{3x-1} \). This type of function is dependent on the argument inside the root being non-negative as even roots are only defined for non-negative numbers in the real numbers.

Step 2: Set the Expression Inside the Root Non-Negative

To determine where the function is continuous, we must set the expression within the root, \( 3x-1 \), to be greater than or equal to zero. This is necessary because even roots like the fourth root are undefined for negative numbers.

Step 3: Solve the Inequality

The inequality \( 3x - 1 \geq 0 \) is solved by adding 1 to both sides, resulting in \( 3x \geq 1 \). Next, divide both sides by 3 to isolate \( x \), giving \( x \geq \frac{1}{3} \). This inequality indicates the set of \( x \)-values for which the function is defined and continuous.

Step 4: Conclusion of Continuity

The function \( y=\sqrt[4]{3x-1} \) is continuous for all \( x \) values satisfying the inequality from Step 3. Therefore, the function is continuous for \( x \geq \frac{1}{3} \).

Key Concepts

Inequalities

Inequalities are mathematical expressions used to show the relationship between two quantities when they are not equal. They use symbols such as "\(<\), "\(\leq\)", "\(>\)", and "\(\geq\)" to indicate different types of comparisons.
Understanding inequalities is crucial in solving real-world problems where values have limits or ranges. In calculus, they help us determine the domain of functions, which is the set of input values for which the functions are defined and behave logically.

For example, in the case of the fourth root function \( y=\sqrt[4]{3x-1} \), we need to ensure the argument, \(3x-1\), stays non-negative. This results in the inequality \(3x-1 \geq 0\). By solving this inequality, we find \(x \geq \frac{1}{3}\), which informs us about the values of \(x\) where the function is continuous.

Function Continuity

Continuity in functions is a key concept in calculus. A function is considered continuous at a certain point if its graph can be drawn without lifting the pen from the paper at that point. In more formal terms, a function \( f(x) \) is continuous at a point \( c \) if the following conditions are met:

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

For deeper understanding, the concept of continuity focuses on the absence of interruptions, jumps, or holes in the function graph.

Applying this to our fourth root function \( y=\sqrt[4]{3x-1} \), it is continuous within the interval determined by the inequality \(x \geq \frac{1}{3}\). This means that within this range, the function has no breaks or undefined points, essentially forming a seamless path.

Fourth Root Function

A fourth root function is a specific type of radical function involving the fourth root of a variable or expression. It is written as \( y = \sqrt[4]{x} \), where the expression under the root must be non-negative for real numbers.

Like all even root functions, the fourth root function has a domain restricted to non-negative numbers to avoid undefined values in the real number system. For \( y=\sqrt[4]{3x-1} \), the expression \(3x-1\) inside the root must be greater than or equal to zero, leading to the inequality \( x \geq \frac{1}{3} \).

• The principal fourth root means looking at the non-negative aspect of the root.
• Graphically, the fourth root function appears as a gradually increasing curve without dips or sharp turns, which contributes to its continuity on the defined interval.

Understanding the properties of such radical functions helps in determining valid intervals for continuity and ensuring that calculations reflect existing valid results.