Problem 69 - Beginning and Intermediate Algebra

Question

Determine the domain of each function. $$h(x)=\frac{10}{x}$$

Step-by-Step Solution

Answer

The domain of the function \(h(x) = \frac{10}{x}\) is \(D(h) = \{x: x \in \mathbb{R}, x \neq 0\}\).

Step 1: Identify restrictions on x-values

For the given function \(h(x) =\frac{10}{x}\), the only restriction to the domain is that the denominator cannot be zero, as division by zero is undefined. In this case, the denominator is simply "x", so we must exclude the value of x that would make the denominator equal to zero.

Step 2: Find the value of x that makes the denominator zero

Set the denominator equal to zero, and then solve for x: \[x = 0\]

Step 3: State the domain of the function, excluding the value found in Step 2

The domain of the function \(h(x) = \frac{10}{x}\) would include all real numbers, except the value we found in step 2 that would make the denominator equal to zero. Therefore, the domain is: \[D(h) = \{x: x \in \mathbb{R}, x \neq 0\}\]

Key Concepts

Rational Functions

Rational functions are expressions that represent the division of two polynomials. The general form of a rational function is \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomial functions, and \( Q(x) eq 0 \) is the denominator. They are widely used in mathematical modeling and data analysis.

• Rational functions illustrate how inputs are mapped to outputs through division.
• They often describe real-world phenomena such as rates, proportions, and probabilities.

Understanding rational functions is crucial because the division involved requires careful consideration of the denominator, since it cannot equal zero. If the denominator were zero, it would cause an undefined situation, leading us to the next topic.

Division by Zero

Division by zero occurs when a number is divided by zero, which is mathematically undefined. When dealing with rational functions, it's pivotal to ensure that the denominator never equals zero. An undefined result happens because:

• Division by zero doesn't produce a defined value in mathematics.
• It interrupts the continuity of functions, leading to gaps or vertical asymptotes in graphs.

To avoid division by zero, identify which values of the independent variable (usually \( x \)) make the denominator zero and exclude them from the domain. For instance, in the function \( h(x) = \frac{10}{x} \), the denominator \( x \) should not be zero, so \( x = 0 \) is excluded from the domain.

Real Numbers

Real numbers encompass all rational and irrational numbers, forming an infinitely large set that includes every point on the number line. The real number system is vital in defining the domain of rational functions, as it suggests:

• All numbers we work with in typical functions are part of the real numbers.
• It includes integers, fractions, and numbers like \( \sqrt{2} \) or \( \pi \).

For the rational function \( h(x) = \frac{10}{x} \), its domain consists of all real numbers except where division by zero happens, thus \( x eq 0 \). This exclusion is necessary because real numbers enable us to define and graph functions by providing a comprehensive universe of possible inputs.

Function Notation

Function notation is a standard way to express functions in mathematics, using the format \( f(x) \) to denote the output of function \( f \) when \( x \) is the input. It's a succinct, efficient way to refer to functions and their domains and ranges without lengthy descriptions.

• Function notation makes it easy to understand which variable is independent and which is dependent.
• It helps quickly identify and communicate relationships between variables.

In the context of rational functions like \( h(x) = \frac{10}{x} \), function notation clarifies which expressions depend on the chosen inputs. The format \( h(x) \) straightforwardly demonstrates that for any input \( x \) (aside from those excluded to avoid division by zero), there is a corresponding output, making the function rule clear and concise.