Problem 59 - 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) \((f \circ f)(x)\). $$f(x)=x^{3}, \quad g(x)=x^{2}+3 x-1$$

Step-by-Step Solution

Answer

Each composition has a domain of all real numbers, \( \mathbb{R} \).

Step 1: Define Compositions

The composition \((f \circ g)(x)\) means substituting \(g(x)\) into \(f(x)\). So, \((f \circ g)(x) = f(g(x))\). Similarly, \((g \circ f)(x) = g(f(x))\) and \((f \circ f)(x) = f(f(x))\).

Step 2: Solve (f ◦ g)(x)

Substitute the expression for \(g(x)\) into \(f(x)\):\[f(g(x)) = f(x^2 + 3x - 1) = (x^2 + 3x - 1)^3.\]The expression depends on \(x\) such that \(g(x)\) is defined. Since \(g(x)\) is a polynomial, it is defined for all real numbers.

Step 3: Solve (g ◦ f)(x)

Substitute the expression for \(f(x)\) into \(g(x)\):\[g(f(x)) = g(x^3) = (x^3)^2 + 3(x^3) - 1 = x^6 + 3x^3 - 1.\]The domain is all real numbers, as polynomials are defined for all real numbers.

Step 4: Solve (f ◦ f)(x)

Substitute the expression for \(f(x)\) into itself:\[f(f(x)) = f(x^3) = (x^3)^3 = x^9.\]This is a cubic power of \(x\), and the domain is all real numbers because polynomials are defined for all \(x\).

Step 5: Identify Domains

Each composition, \((f \circ g)(x)\), \((g \circ f)(x)\), and \((f \circ f)(x)\), is a polynomial expression. Therefore, the domain for each function is all real numbers, \( \mathbb{R} \).

Key Concepts

Domain of a Function

When considering the domain of a function, we are exploring all the possible input values (usually represented by \( x \)) that allow the function to output real numbers, meaning the function expression is valid and doesn't run into mathematical problems such as division by zero or the square root of a negative number.

Composing two functions, like \( (f \circ g)(x) \) or \( (g \circ f)(x) \), involves nesting one function inside the other. To determine the domain of these compositions, it's crucial to find the domain where both functions are properly defined.

• Evaluate the inner function first. Ensure that all inputs (for \( g(x) \) in \( f(g(x)) \), and \( f(x) \) in \( g(f(x)) \)) are valid.
• Check the resulting expression. The domain of the outer function must also be taken into account.

For polynomial functions, like those in the example, domains typically include all real numbers \( \mathbb{R} \). There are no restrictions like sqrt or division to limit the input values.

Polynomial Functions

Polynomial functions are mathematical expressions involving variables (usually \( x \)) raised to whole number powers, with constants multiplied to these terms. They look like this: \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \).

In such functions:

• Each "term" is a part of the function separated by a plus or minus sign.
• The highest power of \( x \) determines its degree. The greater the degree, the more complex the graph.
• Coefficients (like \( a_n, a_{n-1} \)) affect the stretch and direction of the polynomial graph.

Polynomial functions, such as \( f(x) = x^3 \) and \( g(x) = x^2 + 3x - 1 \) from our exercise, have smooth, continuous curves which extend infinitely in both directions on a graph without breaking.

Importantly, polynomials are always defined for all real numbers—meaning you can plug any real number into the polynomial and expect a result. This feature makes them especially significant in calculus and broader mathematics.

Real Numbers

Real numbers are all the numbers you can think of that fill the number line—rational numbers like integers and fractions, and irrational numbers like \( \sqrt{2} \) and \( \pi \).

The set of real numbers, denoted \( \mathbb{R} \), does not include complex numbers, which have a real part and an imaginary part. However, real numbers are quite all-encompassing. They play a fundamental role in algebra, calculus, and all of mathematics.

In function contexts, when we talk about a domain of all real numbers, \( \mathbb{R} \), we mean any value from this set can be used as an input to the function without causing undefined or problematic operations.

Understanding real numbers is crucial because they underpin many mathematical concepts, allowing for continuous functions that can be easily examined and manipulated without restriction. This completeness and functionality make them indispensable in describing polynomial and other functions.