Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Without a graphing calculator, determine the domain and range of the functions. $$f(x)=(x-1)^{3}+4$$

Step-by-Step Solution

Answer

Domain: \( \mathbb{R} \); Range: \( \mathbb{R} \).

Step 1: Identify the Function Type

The given function is \( f(x) = (x-1)^3 + 4 \). This is a cubic function, which is a type of polynomial function.

Step 2: Determine the Domain

For any polynomial function, the domain is all real numbers because polynomials are defined for any real number input. Therefore, the domain of \( f(x) = (x-1)^3 + 4 \) is \( \mathbb{R} \).

Step 3: Analyze the Transformation

The function \( f(x) = (x-1)^3 + 4 \) is a transformation of the basic cubic function \( g(x) = x^3 \). In this case, the function is translated 1 unit to the right and 4 units up. However, these translations do not affect the range of a cubic function.

Step 4: Determine the Range

The basic cubic function \( x^3 \) has a range of all real numbers (\( \mathbb{R} \)). Since the transformations do not change the essential shape of the cubic function, the range of \( f(x) = (x-1)^3 + 4 \) remains all real numbers. Thus, the range is \( \mathbb{R} \).

Key Concepts

Domain

The domain of a function refers to all the possible input values (or 'x' values) that the function can accept. When dealing with polynomial functions, which include cubic functions, the domain is quite simple. Since polynomials are defined for every real number, these functions have a domain that spans all real numbers. For the function given in the problem, \( f(x) = (x-1)^3 + 4 \), the domain is all real numbers, denoted as \( \mathbb{R} \).

• Definition: The set of all possible input values.
• For polynomials: The domain is typically all real numbers \( (\mathbb{R}) \).
• No restrictions: There are no values that make the function undefined.

Consequently, for any polynomial function, you can confidently state the domain as \( \mathbb{R} \) without worrying about divisions by zero or square roots of negative numbers.

Range

The range of a function is all the possible output values (or 'y' values) that the function can produce. For cubic functions like \( f(x) = (x-1)^3 + 4 \), the range is also all real numbers. This is because cubic functions are continuous and extend infinitely in both the positive and negative directions as their input increases or decreases respectively.

• Definition: The set of all possible output values.
• For cubic functions: The range is all real numbers \( (\mathbb{R}) \).
• Transformation: Shifting or translating the function does not change its range.

Even after translating the basic cubic function up or down, left or right, the overall spread of values remains unchanged. Hence, the output can achieve any real number, confirming that the range for \( f(x) = (x-1)^3 + 4 \) is indeed \( \mathbb{R} \).

Polynomial Function

A polynomial function is an important type of function characterized by terms consisting of variables raised to whole number powers and coefficients. Polynomial functions are written in the general form: \( a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \). In these expressions:

• \( a_n, a_{n-1}, \ldots, a_1, a_0 \) are constants (coefficients).
• \( n \) is a non-negative integer representing the highest degree of the polynomial.
• The exponents of the variables are all non-negative integers.

Cubic functions, like the function in our exercise \( f(x) = (x-1)^3 + 4 \), are a specific type of polynomial function with a degree of 3 because the highest power of the variable "\( x \)" is 3. Understanding polynomial functions is crucial because:

• They often serve as fundamental building blocks in various areas of mathematics.
• They offer continuity and smoothness, making them predictable and easy to manipulate.
• They frequently appear in modeling real-world scenarios and solving engineering problems.

Recognizing and analyzing polynomial functions can provide deep insights into their general behavior and properties, such as domain, range, and graph shape.