Problem 57 - College Algebra with Modeling and Visualization

Question

Exercises \(57-72:\) Use the given \(f(x)\) and \(g(x)\) to find each of the following. Identify its domain. $$ \begin{array}{llll} \text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x) \end{array} $$ $$ f(x)=x^{3}, \quad g(x)=x^{2}+3 x-1 $$

Step-by-Step Solution

Answer

All three functions have domain all real numbers \((-\infty, \infty)\).

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

Find the expression for \((f \circ g)(x)\) by substituting \(g(x)\) into \(f(x)\). The function \(f(x) = x^3\) becomes \(f(g(x)) = (g(x))^3 = (x^2 + 3x - 1)^3\).

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

The domain of \((f \circ g)(x)\) is determined by \(g(x) = x^2 + 3x - 1\). Since this is a polynomial, its domain is all real numbers \((-\infty, \infty)\).

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

Find the expression for \((g \circ f)(x)\) by substituting \(f(x)\) into \(g(x)\). The function \(g(x) = x^2 + 3x - 1\) becomes \(g(f(x)) = (x^3)^2 + 3(x^3) - 1 = x^6 + 3x^3 - 1\).

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

The domain of \((g \circ f)(x)\) is determined by \(f(x) = x^3\). Since \(f(x)\) is a polynomial, its domain is all real numbers \((-\infty, \infty)\).

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

Find the expression for \((f \circ f)(x)\) by substituting \(f(x)\) into itself. The function \(f(x) = x^3\) becomes \(f(f(x)) = (x^3)^3 = x^9\).

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

The domain of \((f \circ f)(x)\) is determined by \(f(x) = x^3\). Since \(f(x)\) is a polynomial, the domain is all real numbers \((-\infty, \infty)\).

Key Concepts

Polynomial Functions

Polynomial functions are a type of mathematical expression consisting of terms made up of variables raised to whole number exponents, typically expressed in a form that includes coefficients. These functions are known for being smooth and continuous, meaning they do not have any breaks, jumps, or sharp corners in their graphs. They're fundamental in algebra and appear frequently in various areas of calculus and higher math.

For instance, the function \( f(x) = x^3 \) is a polynomial function. Its highest exponent, or degree, is 3, making it a **cubic polynomial**. Polynomial functions can include numerous terms, like the function \( g(x) = x^2 + 3x - 1 \), which is a quadratic polynomial because the highest exponent is 2.

Some key points to remember about polynomial functions include:

• They have a domain of all real numbers unless otherwise specified.
• Their range can vary depending on the degree and coefficients of the polynomial.
• They generally involve addition, subtraction, and multiplication, but not division by a variable.

Understanding polynomials is essential because they serve as a foundation for more complex mathematics.

Function Domains

When we talk about the domain of a function, we refer to all the possible input values for a given function. In simpler terms, it's the set of all values "x" can take without making the function undefined. Most polynomial functions, like \( f(x) = x^3 \) and \( g(x) = x^2 + 3x - 1 \), have a domain of all real numbers. This is because polynomials are defined and produce real number outputs for any real number input.

For composite functions like \((f \circ g)(x) = (x^2 + 3x - 1)^3\), the domain is determined by the domain of the inside function, which is \( g(x) \) here. Since \( g(x) \) is a polynomial, its domain is also all real numbers, leading to \((f \circ g)(x)\) having the same domain.

Key aspects of function domains include:

• The domain can affect the range and behavior of the function graph.
• For polynomials, unless specified otherwise, the domain is often all real numbers \((-\infty, \infty)\).
• For composite functions like \( (f \circ f)(x) \), we need to consider the domains of both functions involved.

Understanding domains helps ensure functions are evaluated correctly, preventing any potential mishaps due to undefined operations.

Cubic Functions

Cubic functions are a type of polynomial function where the highest exponent of the variable is 3, such as \( f(x) = x^3 \). They are notable in algebra for their ability to model certain kinds of real-world relationships and are characterized by a curve that changes direction up to twice.

A general form of a cubic function is expressed as \( ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants, and \( a eq 0 \). This form may describe various cubic behaviors, including having one real root and possibly two complex roots.

Some important characteristics of cubic functions include:

• They have a function domain of all real numbers \((-\infty, \infty)\).
• Cubic functions can rise and fall abruptly due to changes in direction, quite distinct from the simpler, smoother shapes of quadratic or linear functions.
• Their graphs typically resemble an "S" shape or reversed "3" passing through the origin if in standard form \( y = x^3 \).

Understanding cubic functions is crucial for graphing and solving polynomials, giving insights into more advanced calculus topics.