Problem 14 - 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. $$\bullet (g \circ f)(x)$$ $$\bullet (f \circ g)(x)$$ $$\bullet (f \circ f)(x)$$ $$f(x)=x^{2}-x+1, g(x)=3 x-5$$

Step-by-Step Solution

Answer

(g◦f)(x) = 3x² - 3x - 2; (f◦g)(x) = 9x² - 33x + 31; (f◦f)(x) = x⁴ - 2x³ + 2x² - x + 1. Domains: all are (-∞, ∞).

Step 1: Find Expression for (g◦f)(x)

To find (g \circ f)(x)\, substitute \, f(x) \, into \, g(x). \( g(x) = 3x - 5 \), and \( f(x) = x^2 - x + 1 \). Thus, \( g(f(x)) = 3(x^2 - x + 1) - 5 \). Simplify this: \[ g(f(x)) = 3x^2 - 3x + 3 - 5 = 3x^2 - 3x - 2 \].Domain: The domain of \, f(x) \, is all real numbers, so the domain of \, (g \circ f)(x) \, is also all real numbers: \, (-\infty, \infty).

Step 2: Find Expression for (f◦g)(x)

To find \( (f \circ g)(x) \), substitute \( g(x) \) into \( f(x) \). \( f(x) = x^2 - x + 1 \), and \( g(x) = 3x - 5 \). Thus, \( f(g(x)) = (3x - 5)^2 - (3x - 5) + 1 \). Simplify the expression: \( f(g(x)) = (3x - 5)(3x - 5) - (3x - 5) + 1 \)\( = 9x^2 - 30x + 25 - 3x + 5 + 1 \)\( = 9x^2 - 33x + 31 \).Domain: The domain of \, g(x) \, is all real numbers, so the domain of \, (f \circ g)(x) \, is also all real numbers: \, (-\infty, \infty).

Step 3: Find Expression for (f◦f)(x)

To find \( (f \circ f)(x) \), substitute \( f(x) \) into itself. \( f(x) = x^2 - x + 1 \), so \( f(f(x)) = (x^2 - x + 1)^2 - (x^2 - x + 1) + 1 \). Expand and simplify: \( f(f(x)) = (x^2 - x + 1)^2 = (x^2 - x + 1)(x^2 - x + 1) \)\( = x^4 - 2x^3 + 3x^2 - 2x + 1 \)Now subtract: \( f(f(x)) = x^4 - 2x^3 + 3x^2 - 2x + 1 - x^2 + x - 1 + 1 \)\( = x^4 - 2x^3 + 2x^2 - x + 1 \).Domain: The domain of \, f(x) \, is all real numbers, so the domain of \, (f \circ f)(x) \, is also all real numbers: \, (-\infty, \infty).

Key Concepts

Domain of a Function

In the realm of functions, the term *domain* refers to all the possible input values (usually *x*) that can be plugged into the function without leading to mathematical impossibilities like division by zero or square roots of negative numbers.
For the given functions, both \( f(x) = x^2 - x + 1 \) and \( g(x) = 3x - 5 \), and their compositions \( (g \circ f)(x), (f \circ g)(x), (f \circ f)(x) \), the domains are all real numbers.
This is because they are polynomial functions, which are defined for every real number. Unlike other types of functions, polynomials don’t have restrictions like division by zero or taking even roots of negative numbers.

• When you deal with composite functions like \((g \circ f)(x)\), it's important to verify both the domain of \(f(x)\) and whether \(g(x)\) can accept all possible outputs of \(f(x)\).
• In these exercises, since both functions are polynomials, the domain simply encompasses all real numbers written in interval notation as \((-\infty, \infty)\).

Polynomial Functions

Polynomial functions are a fundamental type of mathematical function. They involve sums and differences of powers of \(x\) that are non-negative integers. In simpler terms, these functions look like expressions with terms such as \(ax^n\), where \(a\) is a constant and \(n\) is a whole number.
For both \(f(x) = x^2 - x + 1\) and \(g(x) = 3x - 5\), you're dealing with polynomial functions.

• **Degree:** The degree of a polynomial is the highest power of \(x\) in the expression. For \(f(x)\), the degree is 2, while for \(g(x)\), it's 1.
• **Structure:** Polynomials can be easily identified because they are sums or differences of monomial terms. For instance, \(f(x)\) consists of three terms: \(x^2, -x, \) and \(+1\).
• **Behavior:** One key trait of polynomial functions is their smooth and continuous graphs, with no breaks, jumps, or holes.

Polynomials are crucial because they model a wide variety of real-world scenarios, and their simplicity makes them a popular choice in algebra and calculus.

Interval Notation

Interval notation is a method of writing the set of values that a variable, like \(x\), can take. In essence, it helps to concisely express the domain or range of a function.

• **Types of Intervals:** Square brackets \([ ]\) denote inclusion of endpoint values – meaning the number is part of the solution. Parentheses \(( )\) indicate that the endpoint is not included.
• **Representation:** When a function's domain includes all real numbers, you represent it as \((-\infty, \infty)\). This notation indicates that \(x\) can be any value from negative infinity to positive infinity.
• **Usage:** Interval notation is especially helpful in clearly communicating solutions and domains, particularly when dealing with inequalities or domains of functions.

Remember, infinity symbols always use parentheses because infinity is not a number that you can "reach," so it's never actually included in a set or domain. Using this notation regularly makes interpreting mathematical concepts smoother and more efficient.