Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the function at each specified value of the independent variable and simplify. $$f(x)=\sqrt{x+8}+2$$ (a) \(f(-4)\) (b) \(f(8)\) (c) \(f(x-8)\)

Step-by-Step Solution

Answer

The evaluated function for the given values are: (a) f(-4) = 4, (b) f(8) = 6, and (c) f(x-8) = \(\sqrt{x} + 2\).

Step 1: Substitution for f(-4)

First, replace the variable \(x\) with the value -4 in the function. The function becomes \(f(-4)=\sqrt{-4+8}+2\).

Step 2: Simplification for f(-4)

Now, simplify the expression inside the square root and the overall equation: \(f(-4)=\sqrt{4}+2 = 2 + 2 = 4\). The first equality comes from simplifying -4+8, the second equality is simply evaluating \(\sqrt{4}\) and the last is summing up the numbers.

Step 3: Substitution for f(8)

Now, replace the variable \(x\) with the value 8 in the function. So the function becomes \(f(8)=\sqrt{8+8}+2\).

Step 4: Simplification for f(8)

Then, simplify the expression inside the square root and the overall equation: \(f(8)=\sqrt{16}+2 = 4 + 2 = 6\). The first equality comes from simplifying 8+8, the second equality is simply evaluating \(\sqrt{16}\), and the last is summing up the numbers.

Step 5: Substitution for f(x-8)

Now, replace the variable \(x\) with the expression \(x-8\) in the function. So, the function becomes \(f(x-8)=\sqrt{(x-8)+8}+2\).

Step 6: Simplification for f(x-8)

Then, simplify the expression inside the square root: \(f(x-8)=\sqrt{x}+2\). This equality comes from simplifying the expression inside the square root, i.e., \((x-8)+8\) which gives \(x\).

Key Concepts

Independent Variable

In mathematics, an independent variable is a quantity that can take on different values freely and isn't dependent on any other variables in the context of the function. In the function given, \(f(x) = \sqrt{x + 8} + 2\), the independent variable is \(x\). This means that for every value you choose for \(x\), you can compute a unique value for \(f(x)\). When solving problems, knowing which variable is independent helps you understand how a function behaves as that variable changes.
For example:

• When \(x = -4\), the function result is determined solely by substituting this value into the expression \(\sqrt{x + 8} + 2\).
• Similarly, when \(x = 8\), we again substitute \(x\) into the same expression to see how the result changes.

So, the independent variable is the pivot of function evaluation, guiding the computation of function outputs.

Square Root Simplification

Simplifying square roots involves making them as simple as possible, removing the square root where appropriate. This process is critical in evaluating functions accurately and in making problems easier to solve. Let's consider the exercise with \(f(x) = \sqrt{x + 8} + 2\):
In step (a) of the solution, after substituting \(x = -4\), we solve\( \sqrt{-4 + 8} = \sqrt{4} = 2 \). Here:

• Initially, \(-4 + 8\) is evaluated to \(4\).
• Then, the square root \(\sqrt{4}\) simplifies to \(2\). This is because \(2^2 = 4\).

Recognizing and applying simplifications to square roots makes complex algebraic expressions more manageable, improving efficiency in calculations.

Substitution Method

The substitution method involves replacing variables with specific values or expressions, which simplifies the process of solving an equation or evaluating a function. In the given function \(f(x) = \sqrt{x + 8} + 2\), substitution is used in different contexts:

• For \(f(-4)\), you substitute \(-4\) for \(x\), resulting in \(f(-4) = \sqrt{-4 + 8} + 2\).
• Similarly, for \(f(8)\), you substitute \(8\) for \(x\), giving us \(\sqrt{8 + 8} + 2\).
• For \(f(x-8)\), you replace \(x\) with \(x-8\), transitioning it to \(\sqrt{(x-8) + 8} + 2\).

This method is a vital tool in function evaluation, making it easier to find the function's output for different scenarios by just changing what you replace \(x\) with.

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and operators (like addition, subtraction, multiplication, division) that together represent a value or set of values. In the function \(f(x) = \sqrt{x + 8} + 2\):
The expression inside the function \(\sqrt{x + 8} + 2\) is an example of an algebraic expression.
Breaking down this expression:

• \(x + 8\) represents a simple algebraic expression within the function, where \(x\) is the independent variable added to 8.
• \(\sqrt{}\) is an operation applied directly to \(x + 8\), affecting how we deal with the function algebraically.

Understanding algebraic expressions is essential, as they form the basis from which equations and functions are constructed, making them an important element in every math problem involving function evaluation.