91Ó°ÊÓ

In mathematics, the domain of a function refers to the set of all possible input values (usually denoted as "x") that will produce a valid output for the function. For rational functions, finding the domain involves identifying all values of "x" that do not result in division by zero, as division by zero is undefined in mathematics.
In the case of the function \( f(x) = \frac{7x}{x-8} \), we start by looking at the denominator \( x - 8 \). We find what value of "x" makes the denominator zero, because these are the values that need to be excluded from the domain. This means setting the denominator equal to zero and solving for "x" ensures we are aware of the restrictions.

• Set denominator \( x - 8 \) equal to zero.
• Solve for "x" to find \( x = 8 \).

The domain of this function is thus all real numbers except "8". We exclude this from our domain set because the function becomes undefined at this point. Therefore, the domain is all real numbers \( \mathbb{R} \) such that \( x eq 8 \).

An expression becomes undefined under certain conditions, primarily involving division by zero in the case of rational functions. When a denominator is zero, it makes the entire fraction undefined because division by zero is not possible in standard arithmetic. This is why identifying these expressions is crucial when determining the domain.
In \( f(x) = \frac{7x}{x-8} \), the expression becomes undefined when \( x \) is such that \( x - 8 = 0 \). Solving \( x - 8 = 0 \) gives us \( x = 8 \), meaning the function is undefined at this point.

• The expression \( \frac{7x}{x-8} \) does not exist when \( x = 8 \) because it results in division by zero.
• Excluding these values from the domain prevents undefined expressions from affecting calculations and descriptions of the function's behavior.

Real numbers are the set of numbers that include both rational numbers (such as fractions and integers) and irrational numbers (like \( \sqrt{2} \) and \( \pi \)). They form the continuum of numbers that we use in most areas of mathematics to describe quantity, measure, and value.
In the context of finding the domain of a function, particularly for rational functions like \( f(x) = \frac{7x}{x-8} \), we often refer to real numbers when stating which "x" values are allowed.

• Real numbers are denoted by \( \mathbb{R} \).
• They include all numbers along the number line, from negative infinity to positive infinity, except where specified.

When declaring the domain of the function \( f(x) \), we say it is all real numbers except where the function is undefined. Hence, for \( \frac{7x}{x-8} \), the domain is all real numbers, \( \mathbb{R} \), except \( x = 8 \). This shows how real numbers provide a comprehensive set of values from which the domain can be determined.