Question

Let \(f(x)=x^4\) and \(g(x)=x^2\). Find \((f g)(x)\). Then evaluate the product when \(x=3\).

Short answer

The product \( (f g)(x) = x^8 \), and when \( x = 3 \), its value is \( 3^8 = 6561 \).

Step-by-step solution

Define the functions

The problem gives two functions, \( f(x) = x^4 \) and \( g(x) = x^2 \). The goal is to find the composite function \( (f g)(x) \) and then evaluate it for \( x = 3 \).

Compose the functions

Find \( (f g)(x) \) by replacing the \( x \) in function \( f \) with the whole function \( g(x) = x^2 \). This gives \( (f g)(x) = (x^2)^4 \). Simplifying that yields \( (f g)(x) = x^8 \). This is the composite function.

Evaluate the function

Now, to evaluate \( (f g)(3) \), replace \( x \) with \( 3 \). So \( (f g)(3) = 3^8 \).

Key concepts

Polynomial Functions

Polynomial functions are algebraic expressions that involve sums of powers of a variable. These powers are non-negative integers, and each term in the function consists of a coefficient multiplied by the variable raised to an exponent.
Polynomials are foundational in algebra because they model a wide variety of phenomena and are used extensively in calculus, number theory, and throughout mathematics.

• A polynomial can be expressed as, for example: \[ P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \]
• Here, \( a_n, a_{n-1}, \ldots, a_0 \) are coefficients and \( n \) is a non-negative integer.

In the given problem, both functions \( f(x) = x^4 \) and \( g(x) = x^2 \) are polynomials. They are composed into another polynomial \((f g)(x) = x^8\). This illustrates that combining polynomials can result in another higher-degree polynomial.

Exponents

Exponents are a way of expressing repeated multiplication of the same number. When a number \( x \) is raised to the power of \( n \), written as \( x^n \), it means multiplying \( x \) by itself \( n \) times.
Understanding how to handle exponents is crucial in simplifying expressions and solving equations.

• The rule \( (x^m)^n = x^{m \cdot n} \) is especially important, as shown in this problem where \((x^2)^4 = x^{2 \cdot 4} = x^8\).

This simplification is essential for finding the resulting polynomial when functions are composed. In our problem, we've used the properties of exponents to simplify \((f g)(x) = (x^2)^4\) into \(x^8\). Recognizing and applying exponent rules can make such problems much more manageable.

Evaluation of Functions

Evaluating functions is the process of finding the output of a function given an input value. It often involves substituting a number for the variable in the expression and simplifying the result.
It's particularly useful for determining specific values within applied problems, analyzing graphs, and checking solutions.

• For example, to evaluate \((f g)(x)\) for \(x = 3\), plug 3 into our function, yielding \((f g)(3) = 3^8\).
• This simplifies to a numeric value: \(3^8 = 6561\).

This step ensures you understand how to work with the function practically. Evaluating functions verifies how they behave at particular points, providing concrete answers derived from algebraic expressions.