Problem 24 - Precalculus Essentials

Question

Determine whether each equation defines y as a function of \(x .\) $$x y-5 y=1$$

Step-by-Step Solution

Answer

The equation \(xy - 5y = 1\) defines y as a function of x, except when \(x = 5\), as it involves division by zero then.

Step 1: Rewrite the Equation

First, attempt to format the equation so \(y\) is alone on one side. Start by dividing every term by \(x\): this would result in \(y - \frac{5y}{x} = \frac{1}{x}\).

Step 2: Group Y Terms

Next, you must group y terms together. Considering that they are both on the same side, \(y(1- \frac{5}{x}) = \frac{1}{x}\).

Step 3: Solving for Y

Solve for \(y\) by dividing both sides by \(1- \frac{5}{x}\); yielding the solved equation for \(y = \frac{1}{x} \div (1- \frac{5}{x})\). Since the equation is in the form \(y = f(x)\), it defines y as a function of x unless \(x = 5\). This ensures for every x-value, there is exactly one y-value.

Key Concepts

Equation Solving

Equation solving means finding values for variables that make the equation true. To determine if an equation defines one variable as a function of another, you usually solve for one of the variables. An equation is a function if, for each input, it produces exactly one output.

In our exercise, we started with the equation \(xy - 5y = 1\). The objective was to see if \(y\) could be expressed solely in terms of \(x\). By dividing each term by \(x\), we simplified the process of isolating \(y\). We then had \(y - \frac{5y}{x} = \frac{1}{x}\).

The key is to group similar terms together. We factored out \(y\) from the left side, resulting in \(y(1- \frac{5}{x}) = \frac{1}{x}\). By dividing both sides by \(1 - \frac{5}{x}\), we solved for \(y\) and obtained \(y = \frac{1/x}{1 - 5/x}\). This confirms the equation is a function as long as the denominator doesn’t equal zero.

Domain and Range

The domain and range of a function inform us about its potential input and output values.

• The domain of a function is the set of all possible input values \(x\). It defines where the function can operate without issues.
• The range of a function is the set of all possible output values \(y\) that result from applying the function to the domain.

For the equation \(y = \frac{1/x}{1 - 5/x}\), we determine the domain by identifying values of \(x\) that make the equation undefined. In particular, the denominator \(1 - 5/x\) must not be zero because division by zero is undefined. This occurs when \(x = 5\).

Thus, the domain of the function is all real numbers except \(x = 5\). Consequently, the range is all real numbers since \(x\) values aren't limited further by other denominators or limits in the equation.

Function Notation

Function notation elegantly represents how one variable depends on another. In function terms, \(y\) is often denoted as \(f(x)\), meaning \(y\) is a function of \(x\). This notation simplifies expressing more complex relationships.

For the exercise, once we derived \(y = \frac{1/x}{1 - 5/x}\), it expresses \(y\) as \(f(x)\). Function notation is beneficial because it makes the relationship between variables clear and consistent.

• Using notation \(f(x)\), you can swiftly understand the structure of a function and evaluate \(y\) for specific \(x\).
• It also helps in communicating ideas universally, allowing anyone versed in function notation to understand the relationship immediately.

Function notation thus plays a pivotal role in higher mathematics by making expressions cleaner and calculations smoother.