Problem 42 - A Graphical Approach to Precalculus with Limits

Question

Without a graphing calculator, determine the domain and range of the functions. $$f(x)=\sqrt[3]{x+7}-10$$

Step-by-Step Solution

Answer

The domain and range of \(f(x) = \sqrt[3]{x+7} - 10\) are both all real numbers.

Step 1: Understanding Domain for Cube Roots

The domain of a function is the set of all possible input values (x-values) that allow the function to work. The function is given by \(f(x) = \sqrt[3]{x+7} - 10\). Since cube roots are defined for all real numbers, there are no restrictions on \(x\) in this case. This means that the domain is all real numbers.

Step 2: Finding the Range for Cube Root Functions

The range of a function is the set of all possible output values (f(x) or y-values). For the function \(f(x) = \sqrt[3]{x+7} - 10\), the expression \(\sqrt[3]{x+7}\) can be any real number because cube roots can result in any real number. Subtracting 10 from any real number (the output of the cube root) is still a real number. Therefore, the range of the function is also all real numbers.

Key Concepts

Domain of a function

The domain of a function is a fundamental concept in precalculus. It refers to all the possible input values (commonly represented as 'x') for which the function is defined and produces real outputs. For some functions, like square roots or fractions, there are specific restrictions that limit the values that 'x' can take. However, when dealing with cube root functions, such as the one given by \[ f(x) = \sqrt[3]{x+7} - 10 \]it's much simpler.

• Cube root functions are unique because they can take both negative and positive inputs while still producing real numbers as outputs.
• This means there's no restriction on 'x', so the domain includes every real number.

This characteristic of cube roots makes such functions very versatile, as they can be applied over the entire number line without exceptions.

Range of a function

Understanding the range of a function is just as crucial as knowing its domain. The range refers to all the possible output values (often represented as 'f(x)' or 'y') that the function can deliver. In the context of cube root functions, like \[ f(x) = \sqrt[3]{x+7} - 10 \]the range tends to be all real numbers.
Here is why:

• The cube root expression \(\sqrt[3]{x+7}\) itself produces any real number, given that the cube root is defined for negative, zero, and positive values of 'x'.
• When you subtract 10 from any real number, it still results in a real number.

Thus, it ensures that whatever 'x' you input, the output will span all real numbers, making the function's range infinite.

Cube root functions

Cube root functions are a fascinating area of mathematics as they are simpler compared to their square root counterparts in terms of domain and range. The basic form of a cube root function is \[ y = \sqrt[3]{x} \]and they can be transformed in various ways, such as translations and reflections.
Key attributes include:

• Domain: All real numbers. This is because you can input any real number, including both negatives and positives, into \(\sqrt[3]{x}\) and still get a valid output.
• Range: Also all real numbers. Since output values can be negative, zero, or positive, it provides a full range across the entire number line.

For example, when you have a function like \[ f(x) = \sqrt[3]{x+7} - 10 \]it illustrates both the freedom of inputs (domain) and outputs (range) that cube root functions offer. The 'x+7' shifts the graph horizontally, while the '-10' moves it vertically down, yet neither transformation affects the universal applicability of the inputs or the outputs.