Question

Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left-continuity at the endpoints. $$f(x)=(2 x-3)^{2 / 3}$$

Short answer

Answer: The function is continuous on the interval $$(-\infty, \infty)$$.

Step-by-step solution

Analyze the function's domain

First, let's look at the function $$f(x) = (2x-3)^{2/3}$$. The domain of this function is any real number because the expression inside the fractional exponent is squared, which will keep all the results positive or non-negative.

Determine the continuity

Since the domain of the function is all real numbers, we can now determine the overall continuity. A function is continuous on an interval if it has no breaks or jumps in its graph on that interval. A function is considered continuous at a point if the function is defined at the point, and the limit exists at the point. Formally, a function is continuous at a point "a" if and only if: 1. $$f(a)$$ is defined. 2. $$\lim_{x \to a^-} f(x)$$ exists. 3. $$\lim_{x \to a^+} f(x)$$ exists. 4. $$\lim_{x \to a} f(x) = f(a)$$ Since the function is defined and has a real value at each point, the left and right limits will also exist and be equal to each other. Therefore, the function is continuous at each point on the real number line.

Determine the continuous intervals

Since the function is continuous at every point on the real number line, it is continuous on the whole interval $$(-\infty, \infty)$$. So, the interval(s) on which the function $$f(x) = (2x-3)^{2/3}$$ is continuous is $$(-\infty, \infty)$$.

Key concepts

Domain of a function

The domain of a function refers to all the values that can be input into the function without causing any undefined or problematic situations. For the function \(f(x) = (2x-3)^{2/3}\), the expression inside is \(2x-3\). If you consider this expression, it can accept any real number as input since there are no restrictions like division by zero or square roots of negative numbers.
The presence of a fractional exponent initially might raise concerns about negative bases, but in our case, the power \(2/3\) simplifies this. Squaring any real number \((x^2)\) results in a non-negative value, handling the base with care against any negative number concerns.
Hence, the domain of this specific function is all real numbers, denoted by \(\mathbb{R}\) or \((-\infty, \infty)\).

• The function accepts all real numbers.
• No restrictions on the input value.
• Expression \(2x-3\) handles any real number gracefully.

Fractional exponents

Fractional exponents are a way to express roots combined with powers. When you see a fractional exponent, it means you're dealing with both roots and powers in a single notation. For the function \(f(x) = (2x-3)^{2/3}\), the exponent \(\frac{2}{3}\) denotes that the expression is both squared and then cube-rooted.
Mathematically, it can be rewritten as \((2x-3)^2\) under the cube root: \[(2x-3)^{2/3} = \sqrt[3]{(2x-3)^2}\]Step-by-step:
1. **Square the function** - Take \(2x-3\) and square the result to eliminate any sign changes that negative inputs might cause. 2. **Cube root the squared value** - Find the cube root of this squared result to manage the overall size of the expression.
Using fractional exponents helps simplify calculations and understand the nature of the function. They enable seamless handling of otherwise challenging expressions, especially when functions interact as in \(f(x)\) here.

• Fractional exponents indicate root and power.
• They simplify complex expressions.
• Useful in calculus and algebra.

Intervals of continuity

When assessing the continuity of a function, we want to determine where the function has no interruptions - basically, where it draws a smooth and unbroken line on the graph.
A function is considered continuous over an interval if it can be traced without lifting the pen. For the function \(f(x) = (2x-3)^{2/3}\), the domain decision as all real numbers implies it's continuous everywhere on the real number line. This means it doesn’t exhibit breaks, holes, or jumps anywhere.
Continuity at a point means:

• The function is defined at that point.
• The left-hand limit as you approach that point exists.
• The right-hand limit as you approach from the other side exists.
• The limit from both sides results in the actual value of the function at that point.

Here, since all these conditions are satisfied across the entire real number line \((-\infty, \infty)\), the function is continuous everywhere.

• No gaps in the graph.
• Smooth function behavior.
• Continuous over all real numbers.