Problem 63 - A Graphical Approach to Precalculus with Limits

Question

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=\sqrt[3]{x^{2}}, x=8$$

Step-by-Step Solution

Answer

The value of \(f(8)\) is 4.

Step 1: Substitute the value of x into the function

Substitute the given value of \(x = 8\) into the function \(f(x) = \sqrt[3]{x^2}\). This gives us:\[ f(8) = \sqrt[3]{8^2} \]

Step 2: Simplify the expression

First, calculate \(8^2\) which gives us 64. Substitute this back into the expression:\[ f(8) = \sqrt[3]{64} \]

Step 3: Evaluate the cube root

Now, find the cube root of 64, which is 4, as \(4^3 = 64\). Thus, we have:\[ f(8) = 4 \]

Key Concepts

Cube Roots

Exploring cube roots is an intriguing mathematical journey, as it involves understanding how numbers relate when used as multiplicands three times. Essentially, the cube root of a number is a value that, when multiplied by itself twice, gives you the original number. In mathematical terms, if \( n^3 = x \), then \( n \) is the cube root of \( x \). For example, the cube root of 64 is 4 because \( 4 \times 4 \times 4 = 64 \).

• Cube roots can be found using different methods like prime factorization or estimation.
• Understanding cube roots helps in solving higher degree polynomials and algebraic expressions.

In computational terms, the cube root is frequently expressed using the notation \( \sqrt[3]{x} \), and this signifies raising \( x \) to the power of \( \frac{1}{3} \). Recognizing this notation is key for evaluating functions involving cube roots.

Substitution in Functions

Substitution in functions is a practical tool in mathematics. It involves replacing variables in a given function with specific values to determine the result. This process is crucial for evaluating functions at particular points. Let's simplify this with a quick example. Suppose you have a function \( f(x) = x + 3 \). If you need to evaluate \( f(2) \), you replace \( x \) with 2, resulting in \( f(2) = 2 + 3 = 5 \).

• This technique simplifies the evaluation process by focusing on one variable at a time.
• It is extensively used in calculus and algebra for function evaluations at specific points.

By consistently practicing substitution, you can develop a deeper understanding and quick reflexes when dealing with more complex functions.

Simplifying Expressions

Simplifying expressions is a fundamental skill that forms the backbone of many mathematical solutions. It involves reducing complex expressions into their simplest form without changing their value. This process can make subsequent calculations much easier and more efficient.

• To simplify an expression, you often combine like terms or use mathematical properties such as distribution.
• For instance, in the expression \( 8^2 \), simplifying by calculating gives \( 64 \).

When you stay organized and follow rules for operations—like handling exponents and roots appropriately—simplifying expressions becomes more intuitive. It becomes easier over time as you familiarize yourself with algebraic rules and patterns, enabling you to tackle problems of increasing complexity.