Problem 66 - A Graphical Approach to College Algebra

Question

Use \(f(x)\) and \(g(x)\) to find each composition. Identify its domain. (Use a calculator if necessary to find the domain.) (a) \((f \circ g)(x) \quad\) (b) \((g \circ f)(x) \quad\) (c) \((f \circ f)(x)\). $$f(x)=\frac{x-3}{2}, \quad g(x)=2 x+3$$

Step-by-Step Solution

Answer

All compositions result in all real numbers as domains.

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

The composition \((f \circ g)(x)\) means \(f(g(x))\). Start by substituting \(g(x) = 2x + 3\) into \(f(x)\):\[f(g(x)) = f(2x + 3) = \frac{(2x + 3) - 3}{2}\].Simplify the expression:\[\frac{2x + 3 - 3}{2} = \frac{2x}{2} = x\].Thus, \((f \circ g)(x) = x\).

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

Since \((f \circ g)(x) = x\) is a linear function, its domain is all real numbers. Therefore, the domain is \(( -\infty, \infty )\).

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

The composition \((g \circ f)(x)\) means \(g(f(x))\). Substitute \(f(x) = \frac{x-3}{2}\) into \(g(x)\):\[g(f(x)) = g\left(\frac{x-3}{2}\right) = 2\left(\frac{x-3}{2}\right) + 3\].Simplify the expression:\[2 \times \frac{x-3}{2} + 3 = (x - 3) + 3 = x\].Thus, \((g \circ f)(x) = x\).

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

Since \((g \circ f)(x) = x\) is a linear function, its domain is all real numbers. Therefore, the domain is \(( -\infty, \infty )\).

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

The composition \((f \circ f)(x)\) means \(f(f(x))\). Substitute \(f(x) = \frac{x-3}{2}\) into itself:\[f(f(x)) = f\left(\frac{x-3}{2}\right) = \frac{\left(\frac{x-3}{2}\right) - 3}{2}\].Simplify the expression:\[\frac{(x-3)-6}{4} = \frac{x - 9}{4}\].Thus, \((f \circ f)(x) = \frac{x - 9}{4}\).

Step 6: Identify the Domain of \((f \circ f)(x)\)

Since \((f \circ f)(x) = \frac{x - 9}{4}\) is a linear function (a polynomial), its domain is all real numbers. Therefore, the domain is \(( -\infty, \infty )\).

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values (typically represented as \(x\)) that the function can accept. When we're dealing with linear and polynomial functions, determining the domain is often straightforward.
A function's domain is crucial because it tells us which inputs can be used without causing errors, such as division by zero or square roots of negative numbers.

For example, if a function is defined as \(f(x) = \frac{x-3}{2}\), to find the domain, check for restrictions like division by zero or undefined expressions. However, in this case, since neither is present, the domain is all real numbers:

• Linear functions: Typically have domains of \(-\infty, \infty\)
• Polynomials: Also generally have domains of \(-\infty, \infty\)

By understanding the domain, you ensure the function works correctly over its entire range.

Linear Function

A linear function is a type of function with a graph that forms a straight line. Its general form is \(f(x) = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. Linear functions have several key characteristics:

• They have a constant rate of change, called the slope \(m\).
• Their graph is always a straight line.
• The domain of a linear function is usually all real numbers \((-\infty, \infty)\).

When functions like \(f(x) = \frac{x-3}{2}\) or \(f(g(x)) = x\) appear, notice they fit the linear form. They don't have restrictions like dividing by zero or square roots, meaning their domain stretches across all real numbers.
Hence, interpreting and graphing linear functions becomes simpler. You can expect predictable output as \(x\) changes—making them easy to understand.

Polynomial Function

Polynomial functions include terms with variables raised to whole number powers. Their general form is \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n, a_{n-1}, \ldots, a_0\) are constants and \(n\) is a non-negative integer.
Characteristics include:

• Curved graphs rather than straight lines.
• A domain that is typically all real numbers \((-\infty, \infty)\).
• A degree that indicates the highest power of the variable, which affects the graph's shape.

Take the example \(f(f(x)) = \frac{x - 9}{4}\), technically a linear polynomial function because it retains a linear form even after composition. Understanding this form helps in recognizing the broader family of polynomial functions.
Although polynomial functions often cover entire domains without restriction, exceptions can arise when dealing with division. However, for pure polynomials, the primary consideration is their general form and how they are graphically depicted.