Problem 30 - College Algebra

Question

In Exercises \(21-32,\) evaluate each function at the given values of the independent variable and simplify. $$f(x)=\frac{4 x^{3}+1}{x^{3}}$$ a. \(f(2)\) b. \(f(-2) \quad\) c. \(f(-x)\)

Step-by-Step Solution

Answer

The solutions are: \(f(2) = 5\), \(f(-2) = -3\), and \(f(-x) = -4-\frac{1}{x^{3}}\)

Step 1: Substitute \(x = 2\)

Substitute \(x = 2\) in the function \(f(x)=\frac{4 x^{3}+1}{x^{3}}\) to get \(f(2)=\frac{4*(2)^{3}+1}{(2)^{3}}\). Perform the necessary arithmetic operations to simplify the resulting expression, which gives: \(f(2)=5\)

Step 2: Substitute \(x = -2\)

Substitute \(x = -2\) in the function \(f(x)=\frac{4 x^{3}+1}{x^{3}}\) to get \(f(-2)=\frac{4*(-2)^{3}+1}{(-2)^{3}}\). Perform the necessary arithmetic operations to simplify the resulting expression, which gives: \(f(-2)=-3\)

Step 3: Substitute \(x = -x\)

Substitute \(x = -x\) in the function \(f(x)=\frac{4 x^{3}+1}{x^{3}}\) to get \(f(-x)=\frac{4*(-x)^{3}+1}{(-x)^{3}}\). This simplifies to \(f(-x)=-4-\frac{1}{x^{3}}\)

Key Concepts

Function Simplification

Understanding how to simplify functions is essential when evaluating them at specific inputs. Simplification typically involves reducing expressions to their simplest form, making calculations easier and more intuitive. Take, for instance, the function presented in the exercise:
\begin{center} \( f(x) = \frac{4x^3 + 1}{x^3} \)\right. \right.\right.This function can be simplified before substituting any values for \(x\). Notice that \(4x^3\) and \(1\) are both divided by \(x^3\), which allows us to separate the expression into \(4 + \frac{1}{x^3}\). This is a crucial step because it shows us that evaluating the function at positive or negative values for \(x\) will have a significant impact due to the cubic term \(x^3\).
Surprisingly, simplifying isn't always about making an expression shorter. Sometimes, expanding or reorganizing terms can clarify how a function behaves for different inputs, aiding the evaluation process. Simplification aims to restructure the function in a way that any substitutions or further operations become straightforward.

Polynomial Functions

Polynomial functions are one of the most fundamental types of functions in algebra. They are composed of terms that are simply variable expressions raised to whole-number exponents added together, like \(x^2\), \(3x\), or \(4\). The function from the exercise, \( 4x^3 \), is a term in a polynomial function.
What characterizes polynomial functions, among other features, is their smooth continuous curves, and the fact that their graphs have no breaks, holes, or sharp corners. This is important for understanding how the function's value changes with different inputs. As seen in our exercise, even though our polynomial is part of a more complex function (a rational expression), analyzing its behavior is indispensable when evaluating the function for positive and negative values of \(x\).
Having the ability to recognize and work with polynomials will help students not only in evaluating functions as in this exercise but also in operations such as division, factoring, and finding roots, crucial skills for advanced algebraic problem-solving.

Rational Expressions

Rational expressions are ratios of two polynomial functions, similar to fractions. In the exercise, the function \( \frac{4x^3 + 1}{x^3} \) is a rational expression because both the numerator and the denominator are polynomials.
These expressions behave much like fractions. They can be simplified by dividing both the numerator and denominator by their greatest common factor. However, with rational expressions, students must be cautious about the domain of the function – that is, the set of all possible input values for \(x\). For instance, if \(x\) were '0', the given function would be undefined as division by zero is not allowed.
When evaluating rational expressions, students must also watch out for the potential simplifications, as shown in the substitutions from the exercise. The outcomes, such as \(f(2) = 5\) and \(f(-2) = -3\), demonstrate that the rational function can yield both positive and negative values. These evaluations are vital for understanding how the function behaves and are foundational in plotting the graph of the expression.