Problem 21 - 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. $$\bullet (g \circ f)(x)$$ $$\bullet (f \circ g)(x)$$ $$\bullet (f \circ f)(x)$$ $$f(x)=3 x-1, g(x)=\frac{1}{x+3}$$

Step-by-Step Solution

Answer

(g \circ f)(x) = \frac{1}{3x + 2}, (f \circ g)(x) = \frac{-x}{x+3}, (f \circ f)(x) = 9x - 4; Domains: (-\infty, -\frac{2}{3}) \cup (-\frac{2}{3}, \infty), (-\infty, -3) \cup (-3, \infty), (-\infty, \infty).

Step 1: Find \((g \, \circ \, f)(x)\)

To find \((g \circ f)(x)\), substitute \(f(x) = 3x - 1\) into \(g(x) = \frac{1}{x+3}\). This yields:\[ g(f(x)) = g(3x - 1) = \frac{1}{(3x - 1) + 3} = \frac{1}{3x + 2} \] The domain of \(g(f(x))\) is all real numbers except where the denominator equals zero, so we solve \(3x + 2 = 0\) to get \(x = -\frac{2}{3}\). Thus, the domain in interval notation is \((-\infty, -\frac{2}{3}) \cup (-\frac{2}{3}, \infty)\).

Step 2: Find \((f \, \circ \, g)(x)\)

To find \((f \circ g)(x)\), substitute \(g(x) = \frac{1}{x+3}\) into \(f(x) = 3x - 1\). This yields:\[ f(g(x)) = f\left(\frac{1}{x+3}\right) = 3\left(\frac{1}{x+3}\right) - 1 = \frac{3}{x+3} - 1 = \frac{3 - (x+3)}{x+3} = \frac{3 - x - 3}{x+3} = \frac{-x}{x+3} \]The domain of \(f(g(x))\) is all real numbers such that \(x + 3 eq 0\), hence \(x eq -3\). The domain in interval notation is \((-\infty, -3) \cup (-3, \infty)\).

Step 3: Find \((f \, \circ \, f)(x)\)

To find \((f \circ f)(x)\), substitute \(f(x) = 3x - 1\) into itself. This yields:\[ f(f(x)) = f(3x - 1) = 3(3x - 1) - 1 = 9x - 3 - 1 = 9x - 4 \]The function \(9x - 4\) is a polynomial, which means the domain is all real numbers \((-\infty, \infty)\).

Key Concepts

Understanding Function Domain

The domain of a function is all the possible values of the input, often represented by the symbol \( x \), for which the function is defined. In simpler terms, the domain tells us what kind of numbers you can safely plug into the function without causing any issues like division by zero or taking the square root of a negative number.

To determine the domain of a function, look at the expressions inside the function. For polynomial functions and linear expressions, the domain is usually all real numbers, ranging from negative to positive infinity, because there are no values of \( x \) that make these expressions undefined. We write this in interval notation as \((-inity, inity)\).

However, functions involving fractions have some restrictions.

• If the function has a denominator that includes \( x \), we exclude the \( x \)-values that make the denominator zero.
• For example, in the function \( g(x) = \frac{1}{x+3} \), you'd set the denominator \( x + 3 \) equal to zero and solve for \( x \), which here gives \( x = -3 \). This means \(-3\) is not included in the domain.

The solution covers finding the domain of composite functions where two functions are combined. They require you to ensure that none of the inputs to the individual functions or their compositions result in an undefined expression.

Exploring Function Notation

Function notation is a way of expressing a function that allows you to see what the actual function is doing. It's written as \( f(x) \), where \( f \) represents the name of the function and \( x \) signifies the variable or the input of the function.

This notation helps you easily substitute values and even whole functions into the equation.

• For example, if \( f(x) = 3x - 1 \) and you want to substitute \( g(x) = \frac{1}{x+3} \) into \( f(x) \), you replace every \( x \) in \( f(x) \) with \( g(x) \).
• This results in \( f(g(x)) = f\left(\frac{1}{x+3}\right) \), a new function which then can be solved and simplified.

Function notation maintains clarity and consistency, ensuring you can work with complicated equations without misunderstanding which operations to perform. It provides a flexible way to deal with single functions or compositions of functions by making clear what substitutions are being performed.

Examining Polynomial Functions

Polynomial functions are a specific type of function that consist of terms called monomials. Each term includes a coefficient and a variable raised to a non-negative integer power. The simplest polynomial functions are linear, like \( f(x) = 3x - 1 \).

These functions are continuous and smooth, with no breaks or sharp turns. The degree of the polynomial ( the highest exponent of \( x \) ) tells us about the number of roots and the overall shape of the graph.

• Because polynomial functions are composed of sums and products of coefficients and powers of \( x \), their domain is all real numbers unless otherwise restricted.
• For example, after finding \( f(f(x)) \) in our exercise, the result \( 9x - 4 \) is another polynomial, so its domain is \((-inity, inity)\) as there are no restrictions involved.

Understanding how to work with polynomial functions is critical in algebra, as they form the basis of many more complex equations and real-world applications.