Problem 64 - 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 \(f(x)\) when \(x = -8\) is 4.

Step 1: Substitute the Given Value

Start by substituting \(x = -8\) into the function \(f(x) = \sqrt[3]{x^2}\). This will give us: \(f(-8) = \sqrt[3]{(-8)^2}\).

Step 2: Simplify Inside the Root

Calculate \((-8)^2\) which is 64. This transforms the expression into \(f(-8) = \sqrt[3]{64}\).

Step 3: Apply Cube Root

Find the cube root of 64. Since \(4^3 = 64\), the cube root of 64 is 4. Therefore, \(f(-8) = 4\).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept whereby you substitute a given value for the variable in the function. It allows us to compute the output value of a specific input. Functions are expressed in the form of equations, such as \(f(x)\). The notation \(f(x)\) indicates that the function is called \(f\) and it depends on the variable \(x\). When evaluating a function, we simply replace \(x\) with the specified number. This is akin to finding the answer to the question "What does this function equal when \(x\) is given a certain value?"

• Identify the function expression.
• Substitute the given number for \(x\).
• Simplify the expression to find the result.

For example, in the function \(f(x) = \sqrt[3]{x^2}\), if we need to find \(f(-8)\), we replace \(x\) with \(-8\), then simplify the expression. This leads us to better understand the behavior of the function at that specific point.

Cube Roots

Cube roots are introduced when calculating a number which, when multiplied by itself three times, gives the original number. The cube root of a number \(y\) is expressed as \(\sqrt[3]{y}\). This concept is essential, especially when tackling polynomial equations or simplifying radicals in calculus and algebra. Cube roots are analogous to square roots, but they operate in the three-dimensional space of numbers.

• Understand the notation: \(\sqrt[3]{64}\) implies "What number, when cubed, equals 64?"
• Recognize common cube roots: Knowing that smaller cube roots, like \(2^3 = 8\) or \(4^3 = 64\), can speed up calculations.
• Solve the cube root step by step: Trial with familiar integers to check correctness.

In our example, finding the cube root of 64 involves recognizing that 4 is the correct cube root because \(4^3 = 64\). Being comfortable with such computations helps in various mathematical contexts.

Order of Operations

Mastering the order of operations is key to solving mathematical problems correctly. Sometimes captured by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), this hierarchy ensures that we approach calculations systematically. The rules are essential for understanding how to simplify complex expressions.

• Start with parentheses: Simplify expressions within parentheses first.
• Handle exponents next: Be it square roots or cube roots, perform these operations early.
• Proceed with multiplication and division: Evaluate from left to right.
• Finish with addition and subtraction: Also from left to right.

In the given problem, the order of operations guides us to first square the number inside the cube root expression, before evaluating the cube root. This methodical approach reduces errors and provides clarity, ensuring accurate results.