Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=\frac{x-3}{2}, g(x)=2 x+3$$

Step-by-Step Solution

Answer

(a) \(x\), domain: \((-\infty, +\infty)\); (b) \(x\), domain: \((-\infty, +\infty)\); (c) \(\frac{x-9}{4}\), domain: \((-\infty, +\infty)\).

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

The composition \((f \circ g)(x)\) means first applying \(g(x)\) and then applying \(f(x)\) to the result. Start by substituting \(g(x)\) into \(f(x)\). \[ f(g(x)) = f(2x + 3) = \frac{(2x + 3) - 3}{2} = \frac{2x}{2} = x \]The expression simplifies to \(x\), so \((f \circ g)(x) = x\).

Step 2: Determine Domain of \((f \circ g)(x)\)

The function \((f \circ g)(x) = x\) is a linear function and has no restrictions. The domain is all real numbers, \((-\infty, +\infty)\).

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

The composition \((g \circ f)(x)\) means applying \(f(x)\) first, then \(g(x)\). Substitute \(f(x)\) into \(g(x)\):\[ g(f(x)) = g\left(\frac{x-3}{2}\right) = 2\left(\frac{x-3}{2}\right) + 3 = (x-3) + 3 = x \]This simplifies to \(x\), so \((g \circ f)(x) = x\).

Step 4: Determine Domain of \((g \circ f)(x)\)

Since \((g \circ f)(x) = x\), it is also a linear function without restrictions. The domain is all real numbers, \((-\infty, +\infty)\).

Step 5: Calculate \((f \circ f)(x)\)

For \((f \circ f)(x)\), substitute \(f(x)\) into itself:\[ f(f(x)) = f\left(\frac{x-3}{2}\right) = \frac{\left(\frac{x-3}{2}\right)-3}{2} \]Simplify the expression:\[ f\left(\frac{x-3}{2}\right) = \frac{x-3-6}{4} = \frac{x-9}{4} \]So, \((f \circ f)(x) = \frac{x-9}{4}\).

Step 6: Determine Domain of \((f \circ f)(x)\)

The function \(\frac{x-9}{4}\) is a linear function without restrictions. The domain is all real numbers, \((-\infty, +\infty)\).

Key Concepts

Domain of a Function

When we talk about the domain of a function, we're discussing the set of all possible inputs (or "x" values) that the function can accept. To determine the domain of a composition of functions, we need to consider the domains of both functions involved. If you're making a composition, remember these steps:

• Identify all "x" values for which both functions are defined.
• Ensure the output from the first function is within the domain of the second function.

In our exercise, we saw that the functions were \(f(x) = \frac{x-3}{2}\) and \(g(x) = 2x + 3\). These are both linear functions, which generally have a domain of all real numbers, \((-\infty, +\infty)\). Since the compositions \((f \circ g)(x)\), \((g \circ f)(x)\), and \((f \circ f)(x)\) resulted in linear expressions, they too have the same unrestricted domain: all real numbers.

Linear Functions

Linear functions are among the simplest types of functions you'll encounter. They have a straightforward form: \(f(x) = mx + b\), where "m" represents the slope, and "b" is the y-intercept. Linear functions create straight lines when graphed, which is why they're often the first functions we learn to graph. These functions have two primary characteristics:

• Their graphs are straight lines.
• They have a constant rate of change, meaning the slope ("m") is the same at any point on the line.

In the context of the exercise, the functions \(f(x) = \frac{x-3}{2}\) and \(g(x) = 2x + 3\) are linear. When composed in various ways, they continued to produce linear functions like \(x\) or \(\frac{x-9}{4}\), maintaining their key characteristics. The domain of these linear functions remains all real numbers, \((-\infty, +\infty)\), due to the absence of any fractions with variables in the denominator or other restrictions.

Function Operations

Function operations include addition, subtraction, multiplication, division, and, importantly for this exercise, composition. Function composition involves plugging the output of one function into the input of another. It's a way to chain functions together and is written using the symbol \((f \circ g)(x)\). When you perform composition, be careful of the order:

• \((f \circ g)(x)\) means apply \(g(x)\) first, then \(f(x)\).
• \((g \circ f)(x)\) means apply \(f(x)\) first, then \(g(x)\).

Understanding composition helps you analyze how different functions transform inputs, one after the other. Our specific exercise demonstrated how the functions \(f(x)\) and \(g(x)\) interact when composed. Each step involved substituting expressions systemically, providing deeper insight into how functions can build upon one another. Function composition is a powerful tool in mathematics, providing flexibility and efficiency in problem-solving processes.