Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the relation represents \(y\) as a function of \(x\). \(y=\frac{1}{x}\)

Step-by-Step Solution

Answer

Yes, \(y = \frac{1}{x}\) is a function of \(x\).

Step 1: Understand the Definition of a Function

A relation represents \(y\) as a function of \(x\) if for every value of \(x\), there is exactly one value of \(y\). This means there cannot be an \(x\) that corresponds to more than one \(y\).

Step 2: Examine the Equation

Consider the equation \(y = \frac{1}{x}\). This equation gives a specific formula for determining \(y\) from any given \(x\).

Step 3: Check Multiple Values of \(x\)

Try different values of \(x\) to ensure that they all yield a single value for \(y\):- For \(x = 2\), \(y = \frac{1}{2}\).- For \(x = -3\), \(y = \frac{1}{-3} = -\frac{1}{3}\).These results indicate that for each \(x\), there is one corresponding \(y\).

Step 4: Consider the Domain Restrictions

Notice that \(x\) cannot be 0, because this would make the denominator zero, which is undefined. However, this does not affect whether \(y\) is a function of \(x\); it merely restricts the domain of the function.

Step 5: Conclusion

Since every value of \(x\) (except \(x = 0\) which is not in the domain) leads to exactly one value of \(y\), the relation \(y = \frac{1}{x}\) is a function of \(x\).

Key Concepts

Domain and Range

In mathematics, understanding the domain and range of a function is essential for analyzing how it behaves. The domain of a function is the set of all possible input values (usually represented by the variable \(x\)) that will produce a valid output. For the function \(y = \frac{1}{x}\), the domain is all real numbers except \(x = 0\), because dividing by zero is undefined.

To find the domain:

• Identify all values of \(x\) for which the function is undefined. Here, it is undefined at \(x = 0\).
• Include all other real numbers as part of the domain.

The range, on the other hand, is the set of all possible outputs \(y\). For \(y = \frac{1}{x}\), the range includes all real numbers except \(y = 0\).

To determine the range:

• Consider the behavior of \(y\) as \(x\) approaches large positive or negative values, resulting in \(y\) approaching zero, but never quite reaching it.
• Recognize that \(y\) can take on positive and negative values, highlighting its complete span over all non-zero real numbers.

Rational Functions

Rational functions are a type of function expressed as the ratio of two polynomials. In simpler terms, they are fractions with polynomials in the numerator and the denominator. The function \(y = \frac{1}{x}\) is a classic example, where \(1\) is a polynomial of degree zero, and \(x\) is a polynomial of degree one.

Important properties of rational functions include:

• Undefined Points: Rational functions are often undefined for values that make the denominator zero, such as \(x = 0\) in \(y = \frac{1}{x}\).
• Asymptotes: These are lines that the graph of a function approaches but never touches. For \(y = \frac{1}{x}\), both vertical and horizontal asymptotes are present at \(x = 0\) and \(y = 0\), respectively.
• Behavior at Infinity: As \(x\) approaches positive or negative infinity, the value of \(y\) aligns closer to zero, indicating that \(y = 0\) is a horizontal asymptote.

Function Notation

Function notation is a convenient way to express relationships between variables in mathematics, typically represented as \(f(x)\). It tells us that \(y\) is a function of \(x\), or that \(f(x)\) represents the value of the function for a given \(x\). In the context of \(y = \frac{1}{x}\), this can be written in function notation as \(f(x) = \frac{1}{x}\).

Key aspects of function notation:

• Clarity and Organization: Function notation helps clearly present the independent variable \((x)\) and the dependent variable \((f(x)\)), enhancing readability.
• Evaluating Functions: To find \(f(x)\) for a specific \(x\), substitute the value into the equation. For instance, \(f(2) = \frac{1}{2}\), and \(f(-3) = -\frac{1}{3}\).
• Understanding Relations: Function notation highlights the one-to-one correspondence between \(x\) and \(y\), essential for defining functions accurately.