Problem 20 - College Algebra

Question

Given \(f(x)=\frac{1}{x},\) and \(g(x)=x-3,\) find the following: a. \(\left(f_{0} g\right)(x)\) b. the domain of \((f \circ g)(x)\) in interval notation c. \((g \circ f)(x)\) d. the domain of \(\left(g_{\circ} f\right)(x)\) e. \(\left(\frac{f}{g}\right) x\)

Step-by-Step Solution

Answer

a. \(\frac{1}{x-3}\); b. \((-\infty, 3) \cup (3, \infty)\); c. \(\frac{1}{x} - 3\); d. \((-\infty, 0) \cup (0, \infty)\); e. \(\frac{1}{x(x-3)}\)

Step 1: Compose the Functions for (f ∘ g)(x)

To find \((f \, \circ \, g)(x)\), substitute \(g(x)=x-3\) into \(f(x)=\frac{1}{x}\). Thus, \((f \, \circ \, g)(x) = f(g(x)) = f(x-3) = \frac{1}{x-3}\).

Step 2: Determine the Domain of (f ∘ g)(x)

The function \((f \, \circ \, g)(x) = \frac{1}{x-3}\) is undefined when its denominator is zero. Set \(x-3=0\) to find this value, which gives \(x=3\). Therefore, the domain of \((f \, \circ \, g)(x)\) in interval notation is \((-\infty, 3) \cup (3, \infty)\).

Step 3: Compose the Functions for (g ∘ f)(x)

To find \((g \circ f)(x)\), substitute \(f(x) = \frac{1}{x}\) into \(g(x)=x-3\). Thus, \((g \circ f)(x) = g(f(x)) = g\left(\frac{1}{x}\right) = \frac{1}{x} - 3\).

Step 4: Determine the Domain of (g ∘ f)(x)

The function \((g \circ f)(x) = \frac{1}{x} - 3\) is undefined when \(x = 0\). Therefore, the domain of \((g \circ f)(x)\) in interval notation is \((-\infty, 0) \cup (0, \infty)\).

Step 5: Calculate the Quotient of Functions (f/g)(x)

The function \((\frac{f}{g})(x)\) is calculated as \(\frac{f(x)}{g(x)} = \frac{\frac{1}{x}}{x-3} = \frac{1}{x(x-3)}\). This expression is undefined for \(x = 0\) and \(x = 3\). Thus, the domain in interval notation is \((-\infty, 0) \cup (0, 3) \cup (3, \infty)\).

Key Concepts

Domain of a Function

Functions are like machines that take an input and produce an output. However, they can't work with every input. The set of inputs that a function can accept is called its domain. Understanding the domain is crucial because it tells us which values make the function work correctly.

To find the domain of a composed function, like \( (f \circ g)(x) \), we look for values that make the function undefined. For instance, \((f \circ g)(x) = \frac{1}{x-3}\) is undefined when the denominator \(x-3\) is zero. Solving \(x-3=0\) gives \(x=3\). Hence, \(x=3\) is not part of the domain. Thus, the domain is all real numbers except 3: \((-\infty, 3) \cup (3, \infty)\).

This process shows why understanding domains is essential, as it reveals the limitations and constraints of the functions involved.

Quotient of Functions

A quotient of functions is a new function created by dividing one function by another. If you have two functions, \(f(x)\) and \(g(x)\), the quotient is denoted \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\). This division is straightforward, but there is an important rule to remember: never divide by zero!

In our example, \(f(x) = \frac{1}{x}\) and \(g(x) = x-3\). Consequently, the quotient is: \[\left(\frac{f}{g}\right)(x) = \frac{\frac{1}{x}}{x - 3} = \frac{1}{x(x - 3)}\]

However, this expression is undefined when either \(x = 0\) or \(x = 3\), since we cannot divide by zero. This affects the domain of \(\left(\frac{f}{g}\right)(x)\), which becomes all real numbers except 0 and 3. In interval notation, the domain is expressed as \((-\infty, 0) \cup (0, 3) \cup (3, \infty)\).

By identifying these restrictions, you ensure that you only evaluate the quotient where it is mathematically valid.

Interval Notation

Interval notation is a way to describe a range of values, showing which numbers are included in the set. It's commonly used for defining the domain of functions. Understanding this notation helps you express complex mathematical ideas simply.

Intervals can be "open" or "closed", depending on whether the endpoints are included. For example:

• Open intervals: \((a, b)\) includes all numbers between \(a\) and \(b\) but not \(a\) and \(b\) themselves.


• Closed intervals: \([a, b]\) includes all numbers between \(a\) and \(b\), as well as \(a\) and \(b\).

Consider the domain of \((f \circ g)(x) = \frac{1}{x - 3}\), which is expressed as \((-\infty, 3) \cup (3, \infty)\). This tells us that all numbers except 3 are included. The symbol \(\cup\) indicates a union of multiple intervals, allowing us to combine sets of numbers.

Mastering interval notation enables you to clearly specify domains and other sets in a concise format, aiding in the clear communication of mathematical ideas.