Problem 59 - Algebra 1

Question

Find the domain of the function. $$y=\frac{3}{2-x}$$

Step-by-Step Solution

Answer

The domain of the function \(y=\frac{3}{2-x}\) is all real numbers except 2.

Step 1: Identify the function

Look at the function \(y=\frac{3}{2-x}\) and recognize that it is a rational function, where the numerator is 3 and the denominator is 2-x.

Step 2: Set Denominator Equal to Zero

Set the denominator equal to zero and solve it. Therefore, set 2 - x = 0. Solving for x gives x = 2.

Step 3: Determine the Domain

Since a function is undefined when the denominator equals zero, the domain consists of all real numbers except the value that makes the denominator zero. In this case, the function is undefined when x = 2. Therefore, the domain of the function \(y=\frac{3}{2-x}\) is all real numbers except 2.

Key Concepts

Rational Function

A rational function is a type of function that is represented by a fraction, with both the numerator and the denominator being polynomials. This type of function takes the form \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) eq 0\).
In the exercise provided, the function is \(y=\frac{3}{2-x}\), which is a rational function.
Here, the numerator is the constant 3, a very simple polynomial (of degree 0), and the denominator is a polynomial \(2-x\), where \(x\) is the variable of the function.

• Key character of rational functions is that their denominators must not be equal to zero.
• This is important because division by zero is undefined in mathematics.

Understanding what constitutes a rational function can help you identify them when tackling mathematical problems.

Undefined Values

When working with rational functions, one important aspect to consider is the values for which the function is undefined. A function is undefined at a particular point if the value results in an expression that is not valid in the mathematical sense.
In rational functions, this usually happens when the denominator equals zero.
Looking back at the function \(y=\frac{3}{2-x}\), if we set the denominator equal to zero, we have:

• \(2-x=0\)
• Solving for \(x\), we find \(x=2\).

This tells us the function is undefined at \(x=2\), meaning that putting \(x=2\) into the function would lead to a division by zero, which is not allowable. Identifying undefined values is crucial for determining the domain of a function and preventing any mathematical mishaps.

Real Numbers

The concept of real numbers is essential in understanding the domain of a function. Real numbers encompass all the possible values that a function can take, including both rational numbers (such as fractions or integers) and irrational numbers (such as the square root of a prime number).
In the context of the exercise, the domain of \(y=\frac{3}{2-x}\) consists of all real numbers except those that make the denominator zero.
This means every real number is acceptable as an input to the function except \(x=2\).

• Real numbers form the broadest category of numbers used to describe quantities that vary continuously, making them crucial in calculus and higher mathematics.
• Think of them as the complete set of numbers that you can plot on a continuous number line without any gaps.

Knowing this helps students comprehend why certain values are restricted in the domain, enabling a more profound understanding of function behavior.