Problem 11 - Precalculus : Building Concepts and Connections

Question

In Exercises \(7-16,\) for the given functions \(f\) and \(g,\) find each composite function and identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \(\left(\frac{f}{g}\right)(x)\) $$f(x)=\frac{1}{x} ; g(x)=\frac{1}{2 x-1}$$

Step-by-Step Solution

Answer

The functions are as follows: a) \((f+g)(x) = \frac{3x-1}{2x^2-x}\) with \(x \neq 0\) and \(x \neq 0.5\), b) \((f-g)(x) = \frac{x-1}{2x^2-x}\) with \(x \neq 0\) and \(x \neq 0.5\), c) \((fg)(x) = \frac{1}{2x^2 - x}\) with \(x \neq 0\) and \(x \neq 0.5\), d) \(\left(\frac{f}{g}\right)(x) = \frac{2x-1}{x}\) with \(x \neq 0\).

Step 1: Find the function \((f+g)(x)\)

For \(f(x)+g(x)\), simply add the two functions together: \[f(x) + g(x) = \frac{1}{x} + \frac{1}{2x-1}\]. To combine these into a single fraction, find a common denominator: \[\frac{1}{x} + \frac{1}{2x-1} = \frac{1(2x-1)+x(1)}{x(2x-1)} = \frac{2x-1+x}{2x^2-x}\] which simplifies to \[\frac{3x-1}{2x^2-x}\]. The domain is all \(x\) such that the denominator does not equal zero. In other words, \(x \neq 0\) and \(x \neq 0.5\).

Step 2: Find the function \((f-g)(x)\)

For \(f(x)-g(x)\), subtract the second function from the first: \[f(x) - g(x) = \frac{1}{x} - \frac{1}{2x-1} = \frac{2x-1-x}{x(2x-1)} = \frac{x-1}{2x^2-x}\]. The domain is all \(x\) such that the denominator does not equal zero. In other words, \(x \neq 0\) and \(x \neq 0.5\).

Step 3: Find the function \((fg)(x)\)

For \((fg)(x)\), multiply the two functions: \[f(x) \cdot g(x) = \left(\frac{1}{x}\right) \cdot \left(\frac{1}{2x-1}\right) = \frac{1}{2x^2 - x}\]. The domain is the set of \(x\) such that the denominator does not equal zero, \(x \neq 0\) and \(x \neq 0.5\).

Step 4: Find the function \(\left(\frac{f}{g}\right)(x)\)

For \(\frac{f}{g}(x)\), divide the first function by the second: \[\frac{f(x)}{g(x)} = \frac{1/x}{1/(2x-1)} = \frac{2x-1}{x}\]. The domain is the set of \(x\) such that \(x \neq 0\).

Key Concepts

Function Operations

When we talk about function operations, we refer to the processes of adding, subtracting, multiplying, or dividing two functions. These operations produce a new function, known as the result function.
To perform these operations, you often begin with two distinct functions, say \(f(x)\) and \(g(x)\). Here are the basic procedures for each operation:

• Addition \((f+g)(x)\): Simply add the function values: \(f(x) + g(x)\).
• Subtraction \((f-g)(x)\): Subtract the second function values from the first: \(f(x) - g(x)\).
• Multiplication \((fg)(x)\): Multiply the functions: \(f(x) \cdot g(x)\).
• Division \(\left(\frac{f}{g}\right)(x)\): Divide the first function by the second: \(\frac{f(x)}{g(x)}\). Note that \(g(x)\) must not be zero for this operation to be valid.

It is essential to work step by step through these operations, simplifying the resulting expressions whenever possible. This often involves finding a common denominator when dealing with rational functions.

Domain of a Function

The domain of a function refers to all the possible input values \(x\) for which the function is defined. It is important to determine this set of values, especially when dealing with composite functions.
Finding the domain requires you to consider the following:

• If the function is given as a fraction, ensure the denominator is not zero.
• Consider any square roots in the expression, as they imply non-negative values under the root for real numbers.
• Check if a logarithm is involved, as it requires positive arguments.

In the context of adding, subtracting, multiplying, or dividing functions like in our example, you determine the domain by combining the domain restrictions of both functions involved.
The resulting domain is the intersection of the domains of \(f\) and \(g\). Sometimes, additional restrictions arise from operations themselves, like division, where the divisor cannot be zero.

Rational Functions

Rational functions are specific types of functions where the function is expressed as the ratio of two polynomials, \(\frac{P(x)}{Q(x)}\). Understanding rational functions involves both recognizing their form and handling their unique characteristics.
Some key features of rational functions include:

• Poles: These are points where the function becomes undefined due to a zero in the denominator. They are significant when determining the domain.
• Asymptotes: These lines a function approaches but never actually reaches. They often occur near the poles, showcasing the function's behavior at extreme values.
• Intercepts: The points where the function crosses the axes, crucial for graphing these functions.

In our exercise, the functions \(f(x) = \frac{1}{x}\) and \(g(x) = \frac{1}{2x-1}\) are both rational. When performing operations like addition or subtraction, it’s common to encounter complex fractions.
Simplifying these involves finding a common denominator, allowing you to combine them effectively into one comprehensive expression. Always be mindful of the domain restrictions that emerge from these simplifications.