Problem 51 - A Graphical Approach to Precalculus with Limits

Question

Use \(f(x)\) and \(g(x)\) to find each composition. Identify is domain. (Use a calculator if necessary to find the domain.) \(\begin{array}{llll}\text { (a) }(f \circ g)(x) & \text { (b) }(g \circ f)(x) & \text { (c) }(f \circ f)(x)\end{array}\) $$f(x)=x^{3}, g(x)=x^{2}+3 x-1$$

Step-by-Step Solution

Answer

All compositions are defined for all real numbers.

Step 1: Understand the Notation

The notation \((f \circ g)(x)\) represents the composition of functions \(f\) and \(g\). It means \(f(g(x))\), or substitute \(g(x)\) into \(f(x)\). Similarly, \((g \circ f)(x)\) means \(g(f(x))\), and \((f \circ f)(x)\) means \(f(f(x))\).

Step 2: Compute \((f \circ g)(x)\)

Substitute \(g(x) = x^2 + 3x - 1\) into \(f(x) = x^3\). This means replacing \(x\) in \(f(x)\) with \(x^2 + 3x - 1\). Thus, \((f \circ g)(x)\) becomes \((x^2 + 3x - 1)^3\).

Step 3: Define Domain for \((f \circ g)(x)\)

Since \((f \circ g)(x) = (x^2 + 3x - 1)^3\) is a polynomial, its domain is all real numbers \((-fty, fty)\).

Step 4: Compute \((g \circ f)(x)\)

Substitute \(f(x) = x^3\) into \(g(x) = x^2 + 3x - 1\). Replace \(x\) in \(g(x)\) with \(x^3\) to get \((g \circ f)(x) = (x^3)^2 + 3(x^3) - 1 = x^6 + 3x^3 - 1\).

Step 5: Define Domain for \((g \circ f)(x)\)

The expression \((g \circ f)(x) = x^6 + 3x^3 - 1\) is also a polynomial, so its domain is all real numbers \((-fty, fty)\).

Step 6: Compute \((f \circ f)(x)\)

Substitute \(f(x) = x^3\) into itself. Replace \(x\) with \(x^3\) in \(f(x)\) resulting in \((f \circ f)(x) = (x^3)^3 = x^9\).

Step 7: Define Domain for \((f \circ f)(x)\)

Since \((f \circ f)(x) = x^9\) is a polynomial, its domain is all real numbers \((-fty, fty)\).

Key Concepts

Polynomial Functions

A polynomial function is a type of mathematical expression that involves sums of powers of a variable. These variables typically have coefficients, which are numbers that multiply the variable.

• The general form of a polynomial is: \(a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), where \(a_n\) to \(a_0\) are coefficients and \(x\) is the variable.
• The degree of the polynomial is the highest power of the variable. For example, in \(x^3 + 2x^2 + x + 5\), the degree is 3.
• Polynomials can be used to model a variety of real-world phenomena in physics, engineering, economics, and more.

Understanding how to manipulate polynomial functions is essential. In composition, as in our problem, we substitute one polynomial into another. This creates a new polynomial with potentially higher degrees.

Domain of a Function

The domain of a function is a set that includes all possible input values (often \(x\) values) that the function can accept without causing any errors.

• For polynomial functions, like the ones we are discussing, the domain is typically all real numbers \((-\infty, \infty)\) because you can substitute any real number into a polynomial and get a real number output.
• However, other function types, such as fractions or roots, may have restrictions because certain inputs can cause undefined operations, like division by zero or taking the square root of a negative number.

In function compositions involving polynomials, the domain remains all real numbers as well because the resulting function after composition is still a polynomial. This simplifies our work when dealing with such expressions.

Notation in Mathematics

Notation in mathematics serves as a language to express complex ideas systematically and succinctly. It is crucial to understand and correctly apply notation to avoid errors in calculations and interpretations.

• Function notation such as \(f(x)\) or \(g(x)\) identifies each function and establishes the rule that tells us what to do with \(x\).
• Composition notation, denoted by \((f \circ g)(x)\), indicates applying one function to the result of another. Specifically, \(f(g(x))\) means plugging the result of function \(g\) into function \(f\).
• Correctly interpreting and using this notation allows us to build complex functions from simpler ones and analyze their behaviors and properties.

Grasping these notations helps in understanding not just composition, but also derivatives, integrals, and other advanced mathematical concepts. Always double-check your notation to ensure clarity and correctness in mathematical communication.