Problem 18 - Intermediate Algebra

Question

Let \(f(x)=2 x+1\) and \(g(x)=x-3 .\) Find each function and give its domain. See Example 1. $$ f / g $$

Step-by-Step Solution

Answer

The function is \( \frac{2x+1}{x-3} \) with domain all real numbers except \( x = 3 \).

Step 1: Understanding the Function Composition

The notation \( f/g \) represents the division of function \( f(x) \) by function \( g(x) \). Specifically, this is equal to \( \frac{f(x)}{g(x)} \). We need to define the resulting function and its domain.

Step 2: Writing the Function Expression

Substitute the given functions into the division notation: \( \frac{f(x)}{g(x)} = \frac{2x + 1}{x - 3} \). This expression represents the division of \( f(x) \) by \( g(x) \).

Step 3: Defining the Domain

The domain of a function is the set of all possible input values (\( x \)) for which the function is defined. Since we're dividing by \( g(x) \), we must ensure that the denominator \( x - 3 eq 0 \). Therefore, the value \( x = 3 \) must be excluded from the domain.

Step 4: Stating the Domain

Given \( g(x) eq 0 \), the domain of \( \frac{f(x)}{g(x)} \) is all real numbers except \( x = 3 \). In interval notation, this is \( (-\infty, 3) \cup (3, \infty) \). This means \( x \) can be any real number except 3.

Key Concepts

Function Composition

When dealing with functions, sometimes we may want to combine them in various ways. One of these ways is through function composition. While the operation in the exercise is division rather than composition, understanding composition helps us get a clearer picture of how functions interact. Function composition involves taking the output of one function and using it as the input for another. If we have two functions, say \( f(x) \) and \( g(x) \), we would denote their composition as \( (f \circ g)(x) = f(g(x)) \). This means you're plugging \( g(x) \) into \( f(x) \).

Here's why composition matters:

• Simplifying Problems: Large problems can often be broken down into a series of smaller functions to handle simpler computations.
• Understanding Processes: Many real-world processes can be viewed as a sequence of function compositions, making them easier to analyze.
• Efficiency: Using compositions can optimize calculations by re-using results, which is beneficial in programming and algorithm design.

While our exercise pertains to using division with functions \( \frac{f(x)}{g(x)} \), it channels a similar principle—using operations to synthesize new functions.

Domain of a Function

The domain of a function is crucial because it tells us all the input values your function can accept. Every function, whether basic like \( y = x+2 \) or complex involving radicals or denominators, has constraints that define its domain.

Domains ensure functions behave properly by avoiding input values that might lead to division by zero or taking square roots of negative numbers.

• No Division by Zero: For expressions like \( \frac{1}{x} \), \( x \) cannot be zero because division by zero is undefined.
• Special Expressions: Functions involving square roots \( \sqrt{x} \) require \( x \geq 0 \) to avoid complex numbers when only real numbers are considered.

In our specific exercise, we are handling a rational function. Here, where \( f(x) = 2x + 1 \) and \( g(x) = x - 3 \), the big concern is ensuring the denominator \( g(x) \) is never zero. Specifically, \( x e 3 \) because it would make \( g(x) = 0 \). Thus, the domain is all real numbers except \( x = 3 \), making the interval notation \( (-\infty, 3) \cup (3, \infty) \). Remember, always look out for values leading to these undefined conditions.

Rational Functions

A rational function is one that can be expressed as the ratio of two polynomials, just like what we see in our exercise where \( \frac{f(x)}{g(x)} = \frac{2x+1}{x-3} \). Rational functions are prevalent because they naturally model situations involving rates and proportions.

These functions have distinct properties:

• Poles: Points where the function is undefined because its denominator is zero. For \( \frac{2x+1}{x-3} \), the pole is at \( x = 3 \).
• Vertical Asymptotes: Near poles, the function can grow extremely large or small, which is depicted by vertical lines at such points, adding graphical significance.
• End Behavior: Analyzing what happens to \( f(x) \) as \( x \) moves to infinity involves understanding horizontal asymptotes, which are the function's long-term trend lines.

In essence, by recognizing a rational function, one acknowledges the interplay of its numerator and denominator, understanding restrictions like its domain, and predicting its graphical behavior. This blend of algebraic and geometric considerations makes rational functions a rich topic in trigonometry and calculus.