Problem 12 - College Algebra and Calculus: An Applied Approach

Question

Find (a) \((f+g)(x)\), (b) \((f-g)(x)\), (c) \((f g)(x)\), and (d) \((f / g)(x)\). What is the domain of \(f / g\) ? \(f(x)=\frac{x}{x+1}, \quad g(x)=x^{3}\)

Step-by-Step Solution

Answer

\((f+g)(x) = \frac{x}{x+1} + x^{3}\), \((f-g)(x) = \frac{x}{x+1} - x^{3}\), \((fg)(x) = \frac{x^{4}}{x+1}\), \((f / g)(x) = \frac{x}{x^{4} + x^{3}}\) for all \(x ≠ 0\), and the domain of \(f / g\) is all real numbers except 0.

Step 1: Calculate \((f+g)(x)\)

To find \((f+g)(x)\), add the expressions for \(f(x)\) and \(g(x)\). So, \((f+g)(x) = f(x) + g(x) = \frac{x}{x+1} + x^{3}\), which cannot be further simplified.

Step 2: Calculate \((f-g)(x)\)

To find \((f-g)(x)\), subtract the expression for \(g(x)\) from the expression for \(f(x)\). So, \((f-g)(x) = f(x) - g(x) = \frac{x}{x+1} - x^{3}\).

Step 3: Compute \((fg)(x)\)

To compute this, multiply the expressions for \(f(x)\) and \(g(x)\), i.e., \((fg)(x) = f(x) . g(x) = \frac{x}{x+1} . x^{3}\). This results in \((fg)(x) = \frac{x^{4}}{x+1}\)

Step 4: Compute \((f / g)(x)\)

To find \((f / g)(x)\), divide the expression for \(f(x)\) by the expression for \(g(x)\). So, \((f / g)(x) = \frac{f(x)}{g(x)} = \frac{x/(x+1)}{x^{3}} = \frac{x}{x^{4} + x^{3}}\). This simplification assumes that \(g(x) ≠ 0\).

Step 5: Establish the domain of \(f / g\)

To establish the domain of \(f / g\), find the values of \(x\) for which \(g(x) ≠ 0\). In other words, where \(x^{3} ≠ 0\), which is true for all \(x ≠ 0\). Thus, the domain of \(f / g\) is all real numbers except 0.

Key Concepts

Algebraic Functions

An algebraic function is a type of function that can be expressed using algebraic operations such as addition, subtraction, multiplication, division, and raising to a finite power. They are composed of polynomials, rational functions, roots, and other algebraic structures. These functions can be graphed on a coordinate plane and often include variables raised to whole numbers or sometimes variables within the denominator.

In the context of the given example, both functions provided, \(f(x) = \frac{x}{x+1}\) and \(g(x) = x^3\), are algebraic functions. The first one, \(f(x)\), is a rational function because it involves the division of two polynomials, while \(g(x)\) is a polynomial function because it is a monomial with a variable raised to a power.

Function Addition and Subtraction

Addition and subtraction of functions is a basic operation in algebra that involves combining functions by adding or subtracting their corresponding outputs for any input value \(x\).

The formula to add two functions \(f\) and \(g\) is simply \( (f+g)(x) = f(x) + g(x) \). Similarly, subtraction is \( (f-g)(x) = f(x) - g(x) \). The operations are performed term by term and can sometimes be simplified, depending on the type of function you're working with. In our example, \( (f+g)(x) = \frac{x}{x+1} + x^3 \) and \( (f-g)(x) = \frac{x}{x+1} - x^3 \) are the expressions resulting from addition and subtraction respectively.

Function Multiplication

Multiplying functions is another important concept in algebra. The product of two functions \(f\) and \(g\) is given by \( (fg)(x) = f(x) \cdot g(x) \).

When multiplying algebraic functions, we perform multiplication across terms just like regular algebraic multiplication. For the given functions, \(fg)(x) = \frac{x}{x+1} \cdot x^3 = \frac{x^4}{x+1}\), indicating that the resulting function is also an algebraic function. It's essential to consider any restrictions on the domain when multiplying, as factors in the denominator cannot be zero.

Function Division

Dividing one function by another, or function division, is the process of creating a quotient function from two given functions \(f\) and \(g\), where \( (f / g)(x) = \frac{f(x)}{g(x)} \), provided \(g(x) ≠ 0\).

In the given exercise, when dividing \(f(x)\) by \(g(x)\), we get the quotient function \( (f / g)(x) = \frac{x / (x+1)}{x^3} = \frac{x}{x^4 + x^3} \). The domain of this new function, which also influences its graph, will exclude values of \(x\) that make \(g(x) = 0\), since division by zero is undefined.

Domain of a Function

The domain of a function is the set of all possible input values \(x\) for which the function is defined. It tells us what input numbers are acceptable, and it's determined by the nature of the algebraic expressions involved.

For instance, when dealing with rational functions or divisions of functions like \( (f / g)(x) \), the domain excludes \(x\) values that lead to division by zero. As shown in the solution, since \(g(x) = x^3\), the function \(g\) is not defined for \(x = 0\). Therefore, the domain of \(f / g\) includes all real numbers except zero. Understanding the domain is critical for graphing functions and for solving algebraic equations.