Problem 14 - Algebra 1: Concepts and Skills

Question

Find the domain of the function. \begin{equation} y=\frac{x}{5} \end{equation}

Step-by-Step Solution

Answer

The domain of the function \( y=\frac{x}{5} \) is all real numbers.

Step 1: Identify the Function

The function given in the problem is \( y=\frac{x}{5} \). Note that x appears in the numerator of a fraction.

Step 2: Analyze for Restrictions

As there is no denominator that could potentially become zero and cause the function to be undefined and no square root over a negative number, there do not seem to be any restrictions on the x-values.

Step 3: Determine the Domain

Since there are no x-values that make the function undefined, the domain includes all real numbers.

Key Concepts

Real numbers

Real numbers are all the numbers you can think of along the number line. They include whole numbers, fractions, and even irrational numbers like \( \pi \) and \( \sqrt{2} \). Basically, anything that isn't imaginary or doesn't have a square root of a negative falls under this category.

• Positive numbers, such as 1, 2, and 3.
• Negative numbers, such as -1, -2, and -3.
• Zero (0).
• Fractions like \( \frac{1}{2} \) and decimal numbers like 3.14.

Thus, for most real-world math problems, when you are asked for the domain, it often involves finding where a function is defined based on these real numbers.

Restrictions on functions

When determining the domain of a function, it is crucial to look for restrictions. These restrictions typically arise in two contexts: where a number causes division by zero or where taking the square root of a negative number is involved.
In the function \( y=\frac{x}{5} \),

• The fraction has a constant denominator of 5, which means it will never reach zero. Thus, no restriction is created here.
• Functions with denominators that can equal zero need a closer look, as these values would make the function undefined at those points.
• Also, watch out for square roots. Negative numbers under a square root in real numbers are not allowed.

In summary, always check if any operation could cause the function to be undefined and thus restrict the values \( x \) can take.

Undefined expressions

Undefined expressions occur when a function doesn’t produce a real number output for an input. Common scenarios leading to undefined expressions include division by zero and square roots of negative numbers.
For the function \( y=\frac{x}{5} \):

• Each operation in the expression is valid across all real numbers since the constant 5 in the denominator never makes the expression undefined.
• Unlike more complex expressions, this function remains defined due to the lack of variables in the denominator or any square root operations.

If a function ends up with an undefined expression based on action within the expression, these spots need identifying. They guide whether any restrictions apply to the domain of the function.