Problem 18 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The expressions are: (a) \((f+g)(x)=\frac{x^4+x^3+x^2+x+1}{x^3(x+1)}\), (b) \((f-g)(x)=\frac{x^4+x^3+x^2+x-1}{x^3(x+1)}\), (c) \((f g)(x)=\frac{x}{x^3(x+1)}\) and (d) \((f/g)(x)=x^4(x+1)\). The domain of \(f/g\) is all real numbers except -1 and 0.

Step 1: Addition of the two functions

The sum of the functions \(f(x)\) and \(g(x)\) is given by \((f+g)(x) = f(x) + g(x) = \frac{x}{x+1} + \frac{1}{x^{3}}.\) To simplify, find a common denominator which in this case is \(x^3(x+1)\), and simplify to get the final expression.

Step 2: Subtraction of the two functions

The difference of the functions \(f(x)\) and \(g(x)\) is given by \((f-g)(x) = f(x) - g(x) = \frac{x}{x+1} - \frac{1}{x^{3}}.\) As in Step 1, find a common denominator which once again is \(x^3(x+1)\), and simplify to obtain the final expression.

Step 3: Multiplication of the two functions

The product of the functions \(f(x)\) and \(g(x)\) is given by \((fg)(x) = f(x) * g(x) = \left(\frac{x}{x+1}\right) * \left(\frac{1}{x^{3}}\right).\) Multiplying these two functions straightforwardly yields the final expression.

Step 4: Division of the two functions

The quotient of the functions \(f(x)\) and \(g(x)\) is given by \((f/g)(x) = f(x) / g(x) = \left(\frac{x}{x+1}\right) / \left(\frac{1}{x^{3}}\right).\) Carrying out this division yields the final expression.

Step 5: Determining the Domain of the division of the functions

The domain of \(f / g\) is all real numbers with the exception of the values which make the denominator equal to zero. As such, solve \(g(x) = 0\) and \(x+1=0\), to find the values which are not part of the domain.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators organized in a structured way to convey some mathematical value. When dealing with function operations such as addition, subtraction, multiplication, or division of functions like in our exercise, understanding algebraic expressions is crucial.

Here, we have two functions, \(f(x) = \frac{x}{x+1}\) and \(g(x) = \frac{1}{x^3}\), that we need to combine in different ways. Each combination will often require:

• Finding a common denominator (for addition and subtraction).
• Multiplying the numerators and denominators (for multiplication).
• Dividing by the reciprocal (for division).

Simplification involves rewriting the expressions in simpler terms, making them easier to understand and work with. A clear understanding of handling algebraic expressions is vital for accurately performing these operations and simplifying the resulting expressions.

Domain Determination

Determining the domain of a function is about figuring out which input values make the function valid and able to produce a real number output. When dealing with rational functions like we have in this exercise, it's important to remember that the denominator cannot be zero, as division by zero is undefined.

For the function \((f/g)(x) = \frac{x}{x+1} / \frac{1}{x^3}\), the domain is restricted by both the denominator in \(f(x)\) and the entire function \((f/g)(x)\).

• \(g(x) = \frac{1}{x^3}\): This is undefined when \(x = 0\).
• \(f(x) = \frac{x}{x+1}\): This is undefined when \(x = -1\), as it makes the denominator zero.

Thus, the domain of \((f/g)(x)\) will exclude \(x = 0\) and \(x = -1\). Therefore, the domain is all real numbers except \(x = 0\) and \(x = -1\).

Rational Functions

Rational functions are fractions that have polynomials in both the numerator and denominator. In this exercise, both functions \(f(x)\) and \(g(x)\) are rational functions. Understanding how to manipulate them is essential when performing function operations or determining the domain.

Key aspects of rational functions to remember include:

• Numerator and denominator expressions: They dictate the behavior of the function.
• Finding common denominations: Crucial for adding or subtracting rational functions.
• Zero values in the denominator: These values are excluded from the domain.

When working with operations involving rational functions, always ensure that any proposed solution does not introduce undefined terms, such as division by zero. Moreover, always simplify results where possible to ensure they are presented in the clearest form.