Problem 61 - A Graphical Approach to College Algebra

Question

Find \(f(x)\) at the indicated value of \(x\). $$f(x)=\sqrt{x^{3}+12}, x=-2$$

Step-by-Step Solution

Answer

The value of \(f(-2)\) is 2.

Step 1: Substitute the Value of x into the Function

Substitute the given value of \(x = -2\) into the function \(f(x) = \sqrt{x^3 + 12}\). This gives us \(f(-2) = \sqrt{(-2)^3 + 12}\).

Step 2: Calculate the Exponentiation

Calculate \((-2)^3\). Since \(-2\) raised to the power of 3 is equal to \(-8\), we have \(f(-2) = \sqrt{-8 + 12}\).

Step 3: Simplify the Expression Inside the Square Root

Add \(-8 + 12\). This simplifies to 4, so we get \(f(-2) = \sqrt{4}\).

Step 4: Evaluate the Square Root

Find the square root of 4, which is 2. Therefore, \(f(-2) = 2\).

Key Concepts

Substitution

Substitution is the process of replacing a variable with its given numerical value. It's like swapping out a letter for a number in an equation. For this exercise, we need to substitute the value of \( x = -2 \) into the function \( f(x) = \sqrt{x^3 + 12} \). When you substitute \(-2\) in place of \(x\), the function changes to \( f(-2) = \sqrt{(-2)^3 + 12} \). When performing substitution, always ensure that every instance of the variable in the function is replaced. This first step ensures that you're evaluating the function specifically for the given value of \(x\).

Exponentiation

Exponentiation involves raising a number to a power. In this context, the task involves finding the cube of \(-2\), written mathematically as \((-2)^3\). Exponentiation is one of the fundamental operations in algebra that helps us understand how numbers grow. When calculating \((-2)^3\), you multiply \(-2\) by itself twice more:

• \((-2) \times (-2) = 4\)
• And then, \(4 \times (-2) = -8\)

This result shows how the sign of the base affects the outcome, producing a negative result when the base is negative and the exponent is odd.

Simplification

Simplification in mathematics is all about making an expression easier to work with. It involves combining like terms and performing elementary operations. After substitution and exponentiation, you arrive at \( \sqrt{-8 + 12} \). Here, simplify by adding the numbers inside the square root:

• \(-8 + 12 = 4\)

Simplification is key to solving equations, as it reduces complexity and makes further calculations more straightforward. In our case, simplifying under the square root gets us closer to the final answer, which is always the goal when solving such problems.

Square Root Calculation

The square root calculation involves finding a number that, when multiplied by itself, equals the number inside the square root. In this problem, after substitution, exponentiation, and simplification, you're left with \(\sqrt{4}\). Finding the square root of 4 is straightforward:

• Since \(2 \times 2 = 4\), the square root of 4 is 2.

Square roots are a common part of algebra that allow us to reverse the process of squaring a number. This final operation in the problem confirms the value of the function at the given \(x\). It is an essential skill for interpreting and solving a variety of mathematical problems.