Problem 13 - Precalculus

Question

Use the given pair of functions to find and simplify expressions for the following functions and state the domain of each using interval notation. $$\bullet (g \circ f)(x)$$ $$\bullet (f \circ g)(x)$$ $$\bullet (f \circ f)(x)$$ $$f(x)=2 x+3, g(x)=x^{2}-9$$

Step-by-Step Solution

Answer

(g ◦ f)(x) = 4x^2 + 12x, (f ◦ g)(x) = 2x^2 - 15, (f ◦ f)(x) = 4x + 9; all domains: (-∞, ∞).

Step 1: Compose f and g (g ◦ f)(x)

To find \((g \circ f)(x)\), substitute \(f(x) = 2x + 3\) into \(g(x)\). So, compute \(g(f(x)) = g(2x+3)\). Since \(g(x) = x^2 - 9\), substitute to get, \(g(2x+3) = (2x+3)^2 - 9\). Thus, expand this as: \((2x+3)^2 = 4x^2 + 12x + 9\), \(g(f(x)) = 4x^2 + 12x + 9 - 9\). Simplify to\((g \circ f)(x) = 4x^2 + 12x\). Domain: Since \(f(x)\) is a linear function and \(g(x)\) is a polynomial, the domain is all real numbers: \((-\infty, \infty)\).

Step 2: Compose g and f (f ◦ g)(x)

To find \((f \circ g)(x)\), substitute \(g(x) = x^2 - 9\) into \(f(x)\). So, compute \(f(g(x)) = f(x^2 - 9)\). Since \(f(x) = 2x + 3\), substitute to get: \(f(x^2 - 9) = 2(x^2 - 9) + 3\). Distribute to get: \(2x^2 - 18 + 3\). Simplify to \((f \circ g)(x) = 2x^2 - 15\). Domain: As \(g(x)\) and \(f(x)\) are both polynomials, the domain is all real numbers: \((-\infty, \infty)\).

Step 3: Compose f with f (f ◦ f)(x)

To find \((f \circ f)(x)\), substitute \(f(x) = 2x + 3\) into \(f(x)\) again. So, compute \(f(f(x)) = f(2x+3)\). Since \(f(x) = 2x + 3\), substitute to get: \(f(2x+3) = 2(2x+3) + 3\). Distribute and simplify to get: \(4x + 6 + 3\). Which simplifies to: \((f \circ f)(x) = 4x + 9\). Domain: The domain of a linear function remains all real numbers, so the domain is \((-\infty, \infty)\).

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial for analyzing mathematical expressions. The domain essentially tells us all the possible input values (also known as the independent variable) for which the function exists.
For most polynomial and linear functions, the domain includes all real numbers. This is generally denoted in interval notation as

• each number from minus infinity to positive infinity is included: \((-\infty, \infty)\).
• Polynomials are continuous, making them defined for any real number,while linear functions also slant without interruptions.

When combining functions through composition, it's appropriate to consider the domain of each involved function separately. Since the given functions, \(f(x)=2x+3\) and \(g(x)=x^2-9\), are a linear and a polynomial respectively, their domains don't contain any restrictions. Whether you're looking at \((g \circ f)(x)\), \((f \circ g)(x)\), or \((f \circ f)(x)\), the domain for all remains \\((-\infty, \infty)\).

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number exponents with coefficients. They form the backbone of much algebra and can range from very simple to highly complex forms.
Here's a look at some essential features:

• They consist of terms added together, each term being a product of a constant and a variable raised to a power.
• The general structure is \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(n\) is a non-negative integer.
• The highest power of the variable indicates the polynomial's degree,which informs us about the general shape and the number of turns in the graph.

In the original exercise, we have \(g(x)=x^2-9\),a quadratic polynomial that opens upwards.Its degree is 2, indicating it's a simple U-shaped curve without real restrictions on domain. Whether isolated or combined with another function, the polynomial maintains its property of being defined for all real numbers.

Linear Functions

Linear functions are one of the foundational building blocks in algebra. They represent the simplest form of relationships between two variables and are characterized by their constant rate of change, also known as the slope.
Key characteristics include:

• The general form of a linear function is \(f(x) = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept.
• The graph of a linear function is a straight line, making them easy to recognize and work with.
• They exhibit a uniform level of change.

In our exercise, \(f(x) = 2x + 3\) is a linear function.It has a slope of 2 and a y-intercept of 3, meaning the line crosses the y-axis at 3 and rises 2 units for every additional unit on the x-axis.Like polynomial functions,linear functions extend infinitely without breaks, leaving their domainas all real numbers \((-\infty, \infty)\).These functions are straightforward in composition and reliable in mathematical calculations.