Problem 13 - College Algebra

Question

Find the domain of each function. $$ h(x)-\frac{4}{\frac{3}{x}-1} $$

Step-by-Step Solution

Answer

The domain of the function \(h(x) = \frac{4}{\frac{3}{x} - 1}\) is (-∞, 3) ∪ (3, +∞).

Step 1: Set the denominator equal to zero

First, set the denominator of the fraction, which is \(\frac{3}{x} - 1\), to zero. It is important to find out when this equals zero, because the function becomes undefined at these points. So, form the equation \(\frac{3}{x} - 1 = 0\).

Step 2: Solve the equation

To find the value of x that makes the equation true, solve \(\frac{3}{x} - 1 = 0\). First, add 1 to both sides of the equation to isolate \(\frac{3}{x}\) on one side: \(\frac{3}{x} = 1\).Then, multiply both sides by x to get:\(3 = x\).

Step 3: Formulate the domain

The domain of the given function consists of all real numbers, except for the value x = 3 which makes the denominator zero. Hence, the domain of the function is specified as (-∞, 3) ∪ (3, +∞).

Key Concepts

Rational Functions

Rational functions are mathematical expressions that involve one polynomial divided by another polynomial. They can be represented generally as \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are both polynomials in terms of variable \( x \). These functions are called 'rational' because they resemble fractions, with a numerator and denominator.

• Rational functions can exhibit a wide range of behavior such as asymptotes, intercepts, and complex graphs.
• The denominator of a rational function cannot equal zero, as division by zero is undefined in mathematics.

This brings us to the next topic: identifying where a rational function becomes "undefined." Recognizing these points is crucial when determining the function's domain.

Undefined Expressions

An expression in mathematics becomes undefined when it involves operations that cannot be performed, such as division by zero. With rational functions, undefined expressions occur when the denominator of the fraction equals zero.
To find these undefined points:

• Set the denominator of the rational function equal to zero.
• Solve the equation to find the values of \( x \) that make the function undefined.

Understanding undefined expressions helps in identifying restrictions on the domain of rational functions. In the provided example, setting the denominator \( \frac{3}{x} - 1 = 0 \) helps figure out that the function is undefined at \( x = 3 \). Thus, \( x = 3 \) is excluded from the domain.

Real Numbers

Real numbers include every number that can be found on the number line. This comprises both rational numbers (such as fractions and integers) and irrational numbers (such as \( \pi \) and \( \sqrt{2} \)).

• Real numbers are involved when defining the domain of functions in general, as they represent all possible values \( x \) can take.
• In our context, the domain of \( h(x) = \frac{4}{\frac{3}{x} - 1} \) involves excluding any real numbers that make the denominator zero, specifically \( x = 3 \).

Identifying all real numbers that a function can accept without becoming undefined is a central part of finding its domain.

Interval Notation

Interval notation is a mathematical method for denoting a set of numbers between two endpoints. It is commonly used to express the domain of a function and clearly indicates which numbers are included or excluded.

• Round brackets \( () \) are used to indicate that an endpoint is not included in the interval (an open interval).
• Square brackets \( [] \) show that an endpoint is included (a closed interval).
• If a function's domain excludes values, such as undefined points, then intervals are combined using the union symbol \( \cup \).

For instance, the domain of the function \( h(x) = \frac{4}{\frac{3}{x} - 1} \) is expressed using interval notation as \( (-\infty, 3) \cup (3, \infty) \), meaning all real numbers except \( x = 3 \). This efficient notation conveys both inclusivity and exclusivity clearly.