Problem 46 - Calculus Early Transcendentals

Question

Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left-continuity at the endpoints. $$f(z)=(z-1)^{3 / 4}$$

Step-by-Step Solution

Answer

Question: Determine the interval(s) of continuity for the function $$f(z) = (z-1)^{3/4}$$. Answer: The function $$f(z) = (z-1)^{3/4}$$ is continuous on the interval $$[1,\infty)$$.

Step 1: Identify the function

We are given the function $$f(z) = (z-1)^{3/4}$$ and asked to find its intervals of continuity.

Step 2: Check the restrictions due to the exponent

The exponent 3/4 consists of two operations: first, taking the cube of the number and then the fourth root. There is no problem when cubing a number, as it is defined for all real numbers. However, the fourth root operation presents restrictions, since the number under the root must be non-negative, i.e., greater than or equal to zero, since taking the fourth root of negative numbers is not defined in the real numbers domain. Let's analyze the expression inside the fourth root: $$(z-1)$$ We need to find \(z\) values that make the expression non-negative. To do so, we can set up the inequality: $$(z-1)\geq0$$

Step 3: Solve the inequality

Solving the inequality, we get: $$(z-1) \geq 0$$ $$z \geq 1$$ This means that for all real numbers $$z$$ greater than or equal to 1, the function $$f(z) = (z-1)^{3/4}$$ is continuous.

Step 4: Write the final answer

The interval(s) on which the function $$f(z) = (z-1)^{3/4}$$ is continuous can be represented as $$[1,\infty)$$.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a variable appears in the exponent. They often look like this: \( a^x \), where \( a \) is a constant base, and \( x \) is the variable. In the original exercise, the function involved a fractional exponent, \((z-1)^{3/4}\). This setup means we deal with both raising to a power and taking a root. Understanding these components is crucial:

• Power: Here, \(3\) signifies cubing the expression \((z-1)\).
• Root: The denominator \(4\) means you need to take the fourth root in the function \((z-1)^{3/4}\), which adds complexity because it restricts the domain to non-negative numbers.

Exponential functions significantly differ from polynomial or linear functions due to their rapid growth or decay, depending on their bases and exponents. They are continuous over their entire domain, but the domain itself might be limited by the need to respect real number operations, as seen with roots. This naturally leads us into discussing inequalities, which help determine when such operations are valid.

Inequalities

Inequalities are mathematical statements that describe a range of values, showing that one side is greater or less than, or equal to, another. In solving for continuity in exponential and other functions, inequalities help us find the domain of the function.

In the exercise, to determine when \((z-1)^{3/4}\) is a valid operation in the real numbers, we set up the inequality \((z-1) \geq 0\). This step ensures that the term under the power is non-negative.

• Solve Inequality: Setting \(z - 1 \geq 0\) gives \(z \geq 1\).
• Interpretation: This tells us the function is defined and continuous for all values \(z\) from 1 onwards, or \([1, \infty)\).

Thus, inequalities are crucial to determining the intervals over which the function is continuous, as they let us establish conditions under which the function's operations adhere to mathematical rules related to the real number system.

Real Numbers

Real numbers encompass all numbers that line up on a number line, including both rational numbers (like fractions) and irrational numbers (like \(\sqrt{2}\)). They are essential in defining the domain of functions, like the one in your exercise.

The function \((z-1)^{3/4}\) is subjected to conditions imposed by the real number system:

• Positive and Zero Values: The base of the fractional exponent \((z-1)\) must be non-negative, since negative bases cannot be taken to even roots in real numbers.
• Range of Continuity: Because of this condition, the function’s domain becomes all real numbers \(z\geq 1\), as negative bases are outside the limits of real number operations here.

Understanding the properties of real numbers helps in identifying valid input values and discussing continuity. This is critical when setting intervals for functions, especially when dealing with operations like roots that impact their applicability.