Problem 43 - Calculus Volume 1

Question

For each pair of functions, find a. \((f \circ g)(x)\) and b. \((g \circ f)(x)\) Simplify the results. Find the domain of each of the results. \(f(x)=x+4, g(x)=4 x-1\)

Step-by-Step Solution

Answer

a. \((f \circ g)(x) = 4x + 3\), Domain: all real numbers b. \((g \circ f)(x) = 4x + 15\), Domain: all real numbers

Step 1: Understanding Composition of Functions

The composition of two functions, say \(f\) and \(g\), denoted as \((f \circ g)(x)\), means that you first apply function \(g\) to \(x\), and then apply function \(f\) to the result of \(g(x)\). Similarly, \((g \circ f)(x)\) means that you first apply \(f\) to \(x\), and then apply \(g\) to the result of \(f(x)\).

Step 2: Composing \((f \circ g)(x)\)

First, substitute \(g(x)\) in place of \(x\) in \(f(x)\). This means finding \((f \circ g)(x) = f(g(x)) = f(4x-1)\). Substitute \(x\) in \(f(x) = x + 4\) with \(4x - 1\): \[f(4x-1) = (4x - 1) + 4 = 4x + 3\].

Step 3: Composing \((g \circ f)(x)\)

Next, substitute \(f(x)\) in place of \(x\) in \(g(x)\). So, find \((g \circ f)(x) = g(f(x)) = g(x+4)\). Substitute \(x\) in \(g(x) = 4x - 1\) with \(x + 4\): \[g(x+4) = 4(x + 4) - 1 = 4x + 16 - 1 = 4x + 15\].

Step 4: Finding Domains

The domain of a function is the set of all possible inputs \(x\). Both \(f\) and \(g\) are defined for all real numbers since they are linear functions. Therefore, the compositions \((f \circ g)(x)\) and \((g \circ f)(x)\) are also defined for all real numbers. The domains are all real numbers \((-fty, fty)\) for both compositions.

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions you will encounter. They have the form \(f(x) = ax + b\), where \(a\) and \(b\) are constants. Linear functions represent straight lines when plotted on a graph. Some key characteristics of linear functions include:

• Slope: The constant \(a\) is the slope, indicating the steepness and direction of the line.
• Y-intercept: The constant \(b\) is the y-intercept, the point where the line crosses the y-axis.
• Graphing: The graph of a linear function is a straight line.

Linear functions are easy to work with because they have a straightforward pattern. They are completely defined by their slope and y-intercept, making calculations less complicated.In the given exercise, the functions \(f(x) = x + 4\) and \(g(x) = 4x - 1\) are examples of linear functions. Their simplicity allows for easy composition, which you'll explore in function composition below. Linear functions also make it easy to determine domains since they are usually defined for all real numbers.

Domain of Functions

The domain of a function is the set of all possible inputs that the function can accept. It is important to determine the domain to understand the scope of the function. For linear functions like \(f(x) = x + 4\) and \(g(x) = 4x - 1\), the domain is all real numbers \((-\infty, \infty)\). This means you can substitute any real number for \(x\) and get a valid output. For composite functions like

• \((f \circ g)(x) = f(g(x)) = 4x + 3\)
• \((g \circ f)(x) = g(f(x)) = 4x + 15\)

the domain remains the same. This is because you are still dealing with linear equations, and there are no restrictions on the values of \(x\) that can be input.Understanding the domain helps ensure that calculations are correct and applicable for all input values, which is particularly straightforward with linear functions.

Mathematical Notation

Mathematical notation is a set of symbols and signs used to represent numbers, functions, and operations in a compact and precise way. Understanding these symbols can make solving problems easier.In the context of function composition, notation plays a crucial role:

• The symbol \((f \circ g)(x)\) represents the composition of functions, indicating that you apply \(g(x)\) first, then \(f\).
• The notation \(f(g(x))\) shows that you substitute the entire function \(g(x)\) in place of \(x\) in \(f\).
• Similarly, \((g \circ f)(x)\) means applying \(f(x)\) first, then \(g\), which is represented as \(g(f(x))\).

Without proper knowledge of these notations, it could be challenging to follow or implement function composition. Notations such as \(f(x) = ax + b\) for linear functions, and interval notation like \((-\infty, \infty)\) for domains, are also important. These compact forms make mathematical communication effective and consistent. Developing familiarity with these symbols is essential for mastering mathematics concepts.