Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the equation represents \(y\) as a function of \(x.\) $$y=\sqrt{x^{2}-1}$$

Step-by-Step Solution

Answer

Yes, the given equation represents \(y\) as a function of \(x\) under the condition that \(x \leq -1\) or \(x \geq 1\).

Step 1: Understanding the concept

First, let's review the concept of a function. A function is a rule that matches an input to an output. In this case, each input \(x\) should correspond to one output \(y\). We'll also need to note how the square root function works: the square root of a number is either positive or 0.

Step 2: Examining the square root

In the formula \(y=\sqrt{x^{2}-1}\), the expression under the square root, \(x^{2}-1\) is squared, meaning it can be positive or 0. But since it is subtracted by 1, it can also be negative.

Step 3: Considering the number of outputs for each input

The output of the square root function is only positive, due to its nature. Therefore, for any value of \(x\), the equation can only produce one value of \(y\), if any. In order to avoid undefined results (such as the square root of a negative number), the domain of \(x\) should be \(x \leq -1\) or \(x \geq 1\).

Step 4: Determine whether the equation represents \(y\) as a function of \(x\)

From our analysis, it is clear that, for every value of \(x\) within the appropriate domain, there is only one corresponding \(y\) value. As such, \(y\) can be said to be a function of \(x\) in the given equation.

Key Concepts

Domain

To better understand the domain of a function, let's unpack what this term really means. The domain of a function is the complete set of possible values for the independent variable, often denoted as "x". Essentially, it tells us which values "x" can take without causing any mathematical mishaps, like division by zero or taking the square root of a negative number.

In the equation of our exercise, where we have the expression \( y = \sqrt{x^{2} - 1} \), we need to be mindful of the fact that you can't take the square root of a negative value without venturing into complex numbers, which aren't part of real-valued functions. Therefore, the expression \( x^2 - 1 \) must be non-negative.

• To ensure it's non-negative, \( x^{2} - 1 \geq 0 \).
• Solving this inequality gives us \( x \leq -1 \) or \( x \geq 1 \).

This tells us that the domain of the function is all real numbers less than or equal to -1, and greater than or equal to 1. For values in this range, the function is well-defined and will yield real outputs.

Square Root Function

The square root function is often represented as \( \sqrt{x} \), and it denotes a number which, when multiplied by itself, gives \( x \). It's important to note that, by convention, the square root function yields only non-negative results. That's why \( \sqrt{4} = 2 \) and not -2, although both numbers squared would indeed yield 4.

In our given problem, the equation is \( y = \sqrt{x^{2} - 1} \). We're dealing with a more complicated expression under the square root. The output of this function, \( y \), is constrained to be non-negative since square root values can't be less than zero in the real number system.

• This constraint limits our possible values of \( y \) to only 0 and positive numbers.
• When \( x^{2} \leq 1 \), \( x^2 - 1 \) becomes negative, leading to an undefined \( y \), emphasizing the importance of a correct domain.

The square root function’s behavior, especially when coupled with expressions like \( x^{2} - 1 \), underscores why understanding both domains and outputs are crucial to deciphering the nature of the function.

Input and Output

In any function, inputs and outputs are kid scenarios as they help us determine the relationship between the variables involved. For our function \( y = \sqrt{x^{2} - 1} \), \( x \) serves as the input and \( y \) as the output.

Understanding inputs and outputs:

• **Input (\( x \))**: These are the values you substitute into the function. Based on our domain discussion, viable inputs are any \( x \leq -1 \) or \( x \geq 1 \).
• **Output (\( y \))**: These are the results you get after performing the function operation. Here, \( y \) can be any non-negative value determined by the square root calculation.

Each valid input (\( x \)) results in exactly one output (\( y \)). This one-to-one relationship confirms \( y \) is indeed a function of \( x \). Hence, they play harmonious roles in verifying that functions are well-defined and behave predictably within their prescribed conditions.