Question

Determine the interval(s) on which the following functions are continuous. Be sure to consider right- and left-continuity at the endpoints. $$f(x)=\sqrt{2 x^{2}-16}$$

Short answer

Answer: The function $$f(x) = \sqrt{2x^2 - 16}$$ is continuous on the intervals $$(-\infty, -2] \cup [2, \infty).$$

Step-by-step solution

Write down the given function

The function is given by $$f(x) = \sqrt{2x^2 - 16}$$

Find the domain of the function

For the function to be continuous, the expression inside the square root must be non-negative (greater than or equal to 0). So, we need to find the values of x for which: $$2x^2 - 16 \ge 0$$

Solve the inequality

To solve the inequality, we can factor the expression to find the values of x for which the inequality holds: $$(2x^2 - 16) \ge 0 \Rightarrow 2(x^2 - 8) \ge 0 \Rightarrow 2(x+2)(x-2) \ge 0$$ Now, we can consider each factor and its sign, noting that when two negatives are multiplied together, the result is positive. In this case, the function will be continuous where: $$x\in (-\infty, -2] \cup [2, \infty)$$

Consider the right- and left-continuity at the endpoints

At the left endpoint x = -2, the function is left-continuous because the square root approaches 0 as x approaches -2 from the left. At the right endpoint x = 2, the function is right-continuous because the square root approaches 0 as x approaches 2 from the right.

Write the interval(s) of continuity

With all the information we have, we can now express the interval(s) of continuity for the function in interval notation: $$f(x) \text{ is continuous on } (-\infty, -2] \cup [2, \infty)$$

Key concepts

Domain of a Function

Understanding the domain of a function is crucial when working with any type of function. It defines all the possible input values (usually represented as "x") for which the function is defined. Put simply, it tells us where a function "works." For the given function \(f(x) = \sqrt{2x^2 - 16}\), finding its domain involves looking at the expression inside the square root.
The expression inside the square root must be non-negative because the square root of a negative number isn't real (in the realm of real numbers).
\[ 2x^2 - 16 \geq 0 \]
To find the domain, solve this inequality to identify the x-values that make the expression valid.
After simplifying, we factor the inequality:
\[ 2(x^2 - 8) \geq 0 \]
And further to:
\[ 2(x + 2)(x - 2) \geq 0 \]
From here, find where \(2(x + 2)(x - 2)\) is greater than or equal to zero. The solution is that x can be any value in the interval \((-\infty, -2]\) or \([2, \infty)\).
This understanding of domain is vital as it dictates where our function can be continuous.

Square Root Function

The square root function is one of the simplest yet most fascinating functions encountered in algebra. The function is noted by the radical symbol \(\sqrt{}\). It gives us a primary component which is critical when understanding the function's domain and continuity.
In our function \(f(x) = \sqrt{2x^2 - 16}\), the square root dictates the possible x values (domain) as the expression inside must stay non-negative.
This is due to the square root producing real numbers only when the operand is zero or positive.

• If the expression inside the square root is zero, \(\sqrt{0} = 0\), which is valid.
• If the expression inside the square root is positive, the square root returns a real positive number, which is also valid.
• If the expression inside is negative, it remains undefined for real numbers.

Understanding these rules ensures that when the square root is applied, the results will produce real numbers, allowing the function to remain continuous within its domain.

Interval Notation

Interval notation is a system used to describe a set of numbers as an interval, showing the start and end points. It's a shorthand way to express continuity, domain, and range of functions.
For our function \(f(x) = \sqrt{2x^2 - 16}\), interval notation helps us neatly capture the valid x-values.

• The notation uses parentheses \(()\) for numbers that aren't included in the interval and brackets \([]\) for numbers that are included.
• The union symbol \(\cup\) connects multiple intervals.

For this function, the interval notation indicating where the function is continuous is:
\((-\infty, -2] \cup [2, \infty)\).
This displays that the function is continuous from negative infinity to -2, including -2, and from 2 to positive infinity, including 2.
Interval notation allows one to communicate these sets quickly and efficiently, ensuring clarity when describing solutions or domains.