Problem 73 - College Algebra Essentials

Question

find and simplify the difference quotient $$ \frac{f(x+h)-f(x)}{h}, h \neq 0 $$ for the given function. $$ f(x)=\frac{1}{x} $$

Step-by-Step Solution

Answer

The simplified form of the difference quotient for the function \(f(x) = \frac{1}{x}\) is \(\frac{-1}{(x)(x+h)}\).

Step 1: Substitute \(f(x+h)\) and \(f(x)\) into the difference quotient

Substitute \(f(x + h)\) and \(f(x)\) into the difference quotient. This results in: \(\frac{1/(x+h) - 1/x}{h}\)

Step 2: Simplify the numerator

To simplify the numerator, find a common denominator. Multiply the first fraction by \(\frac{x}{x}\) and the second fraction by \(\frac{(x + h)}{(x + h)}\). We get: \(\frac{x - (x + h)}{h(x)(x + h)}\). Simplifying this further gives us \(\frac{-h}{h(x)(x + h)}\). We can remove the h from the numerator and denominator giving us: \(\frac{-1}{(x)(x + h)}\)

Step 3: Final simplification of the difference quotient

The final simplification of the difference quotient gives us: \(\frac{-1}{(x)(x+h)}\) which is the simplified form of the difference quotient for the function \(f(x) = \frac{1}{x}\).

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in algebra that involves reducing expressions to their simplest forms. In the context of the difference quotient, this process often involves manipulating fractions and expressions to make them more manageable or readable. For example:

• Identifying like terms and combining them
• Finding common denominators in fractions
• Cancelling terms that appear in both the numerator and the denominator

When simplifying the difference quotient, such as \(\frac{f(x+h) - f(x)}{h}\), we first focus on simplifying the numerator \(f(x+h) - f(x)\). For the function \(f(x)=\frac{1}{x}\), substituting \(f(x+h)\) and \(f(x)\) gives us \(\frac{1}{x+h} - \frac{1}{x}\). We simplify this by finding the least common denominator (LCD), which is \(x(x+h)\), allowing us to subtract the fractions easily. The result at this stage is typically a more complex rational expression that can be further simplified by canceling out terms, ensuring the expression is as reduced as possible. In our example, the difference quotient finally simplifies to \(-\frac{1}{x(x+h)}\). This indicates a clean expression free of extraneous terms, which is often the goal in simplification.

Functions and Their Properties

Understanding functions and their properties is crucial when dealing with expressions like the difference quotient. A function is a rule that relates each element from one set (called the domain) to exactly one element in another set (called the range). For instance, the function \(f(x) = \frac{1}{x}\) assigns the reciprocal to each number \(x\), provided \(x eq 0\).The difference quotient, \(\frac{f(x+h) - f(x)}{h}\), is particularly useful in calculus as it provides the average rate of change of the function \(f(x)\) over a small interval \(h\). This oftentimes approximates the derivative of a function, which describes how the function changes at a specific point. By examining the difference quotient as \(h\) approaches zero, one can assess the function's behavior and continuity. Functions have properties such as domain, range, and asymptotic behavior, all important when evaluating expressions like the difference quotient. In the case of \(f(x) = \frac{1}{x}\), the function has a vertical asymptote at \(x = 0\) since the denominator cannot be zero. Recognizing these properties is vital when analyzing the quotient, ensuring it is evaluated within the acceptable domain.

Rational Expressions

Rational expressions, such as \(\frac{1}{x}\), are fractions where the numerator and/or the denominator are polynomials. Simplifying rational expressions is an essential component of algebra and calculus. This involves:

• Finding a common denominator to combine terms
• Canceling common factors in the numerator and denominator
• Understanding restrictions on the variable (such as values that would make the denominator zero)

In the difference quotient \(\frac{\frac{1}{x+h} - \frac{1}{x}}{h}\), simplifying involves creating a single fraction in the numerator by finding the least common denominator, which simplifies the rational expression into \(\frac{-h}{h(x)(x+h)}\). After removing \(h\) from the numerator and denominator, the expression simplifies to \(-\frac{1}{x(x+h)}\), highlighting that the quotient depends on these variables alone.Rational expressions require careful attention since operations within them are governed by the rules of arithmetic alongside algebraic identities. Mastery of simplifying these expressions allows for greater flexibility and understanding in broader mathematical applications, as it lays the groundwork for solving complex equations and integrating functions in calculus.