Problem 21 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Find \(f(x)\) for the indicated values of \(x\), if possible. (b) Find the domain of \(f\). $$ f(x)=x^{3} \text { for } x=-2,5 $$

Step-by-Step Solution

Answer

f(-2) = -8, f(5) = 125; domain is all real numbers.

Step 1: Evaluate f(x) at x = -2

First, we need to evaluate the function \( f(x) = x^3 \) at \( x = -2 \). Substitute \( x = -2 \) into the function and calculate:\[ f(-2) = (-2)^3 = -8. \]

Step 2: Evaluate f(x) at x = 5

Next, we evaluate the function at \( x = 5 \). Substitute \( x = 5 \) into the function and calculate:\[ f(5) = 5^3 = 125. \]

Step 3: Determine the domain of f(x) = x^3

The function \( f(x) = x^3 \) is a polynomial, and polynomials are defined for all real numbers. Therefore, the domain of \( f(x) \) is \( (-\infty, \infty) \) or all real numbers.

Key Concepts

Domain of a Function

When we talk about the domain of a function, we refer to all the possible input values, usually represented by the variable \(x\), that can be used in the function without causing any issues, such as division by zero or taking the square root of a negative number. In simple words, the domain specifies where the function "works" without problems.
For polynomial functions, like \(f(x) = x^3\), calculating the domain is often straightforward. Polynomial functions are continuous and smooth, with no breaks or undefined values. They can accept any real number as an input value, making their domain all real numbers, which is expressed in interval notation as \((-\infty, \infty)\). This property of polynomial functions is because they are essentially a combination of terms of the form \(ax^n\), where \(a\) is a constant and \(n\) is a non-negative integer.
Therefore, if ever in doubt about the domain of a polynomial, remember it's always all real numbers unless specified otherwise by additional restrictions.

Evaluation of Functions

Evaluating a function means finding the output value when a specific input value is plugged into the function equation. This is a fundamental skill in mathematics, used to understand how functions work and predict their behavior for given inputs.
Here's how you evaluate a function using the example \( f(x) = x^3 \):

• To evaluate \( f(x) \) at \( x = -2 \), we substitute \(-2\) in place of \(x\) in the function: \( f(-2) = (-2)^3 = -8 \).
• Similarly, for \( x = 5 \), substitute \(5\) into the function: \( f(5) = 5^3 = 125 \).
  

The process involves replacing \(x\) with the given value each time and then performing the arithmetic operations according to the function's definition. This gives us the corresponding output or image of the function for that input.
Remember, during evaluation, follow the order of operations (parentheses, exponents, multiplication/division, addition/subtraction) to ensure accuracy.

Real Numbers

Understanding real numbers is crucial in grasping the domain and evaluation of functions. Real numbers include all the numbers that are not imaginary; they encompass all the integers, fractions (or rational numbers), and numbers with non-repeating decimals (or irrational numbers).
Real numbers form a continuous line of numbers extending without end in both positive and negative directions, represented mathematically from negative infinity to positive infinity. This completeness of real numbers gives functions like polynomials a broad domain.
In the context of polynomial functions, knowing that the domain is composed of real numbers means the function can accept:

• Whole numbers such as \(..., -2, -1, 0, 1, 2, ...\)
• Fractions like \(\frac{1}{2}\) or \(-\frac{3}{4}\)
• Irrational numbers such as \(\pi\) or \(\sqrt{2}\)

This wide acceptance makes polynomial functions versatile and applicable across various mathematical problems and real-world situations.
Whether plotting a graph or solving real-life problems, knowing the role of real numbers helps highlight where a function is valid, thus reinforcing your understanding of mathematical concepts.