Problem 89 - Algebra A Combined Function

Question

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8}\), find each function value. $$ f(2) $$

Step-by-Step Solution

Answer

The value of \( f(2) \) is \( \sqrt{7} \).

Step 1: Identify the Function

We are given the function \( f(x) = \sqrt{2x + 3} \). Our task is to find the value of \( f(2) \).

Step 2: Substitute the Value of x

Replace \( x \) with 2 in the function formula: \( f(2) = \sqrt{2(2) + 3} \).

Step 3: Simplify Inside the Square Root

Calculate the expression inside the square root: \( 2 \times 2 + 3 = 4 + 3 = 7 \).

Step 4: Calculate the Square Root

Find the square root of the result: \( \sqrt{7} \).

Step 5: State the Function Value

The function value \( f(2) \) is \( \sqrt{7} \).

Key Concepts

Square Root Function

The square root function is quite essential in mathematics for it calculates the principal square root of a given number or expression. In mathematical terms, the square root of a non-negative number \( x \) is a number \( y \) such that \( y^2 = x \). For example, \( \sqrt{9} = 3 \) because \( 3^2 = 9 \). This function is often represented by the radical symbol \( \sqrt{\ } \), which means 'take the square root'.

• Symbol: \( \sqrt{\ } \)
• Operation: Finding two identical factors of a number
• Requirement: Works with non-negative numbers

In our exercise, we deal with a square root function expressed as \( f(x) = \sqrt{2x + 3} \). This formula suggests we first perform the arithmetic inside the radical before finally extracting the square root.
Remember that the square root function is only defined for values that result in non-negative expressions under the radical.

Cubed Root Function

The cubed root function involves finding a number that, when multiplied by itself three times, gives the original number. Mathematically, for a number \( x \), its cubed root \( \sqrt[3]{x} \) corresponds to a number \( y \) for which \( y^3 = x \). Unlike the square root, the cubed root can work with both positive and negative numbers.

• Symbol: \( \sqrt[3]{\ } \)
• Operation: Finding a factor repeated three times to get the original number
• Flexibility: Can handle both positive and negative numbers

In the bigger picture of our exercise, the concept of the cubed root function is presented with the function \( g(x) = \sqrt[3]{x - 8} \). This makes the cubed root vital in identifying values that satisfy specific conditions set by mathematical functions involving cubes.

Substitution in Functions

Substitution is a straightforward method used to evaluate functions, where we replace the variable in the function's formula with a specific value. This process lets us solve for the function value at that point. It's as if you are dissecting the problem by changing the variable into a numerical part, providing a simpler way to understand the function.

• Process: Replacing variables with specific values
• Purpose: Evaluate and simplify functions at given points

In this exercise, we see substitution in action when we replace \( x \) in \( f(x) = \sqrt{2x + 3} \) with 2, resulting in the expression \( f(2) = \sqrt{2(2) + 3} \). This substitution is crucial in function evaluation as it directly impacts the result's accuracy and understanding.

Simplifying Expressions

Simplifying expressions is an important skill in mathematics, used to streamline complex problems to reach a solution more efficiently. Simplification involves basic arithmetic operations that transform an expression into its simplest form.

• Methods: Applying arithmetic operations like addition, subtraction, multiplication, and division
• Goal: Make expressions more manageable and easy to work with

In our exercise, we encounter simplification when evaluating the expression inside the square root: \( 2 \times 2 + 3 \) simplifies to \( 4 + 3 = 7 \). Simplification helps us to see clearer and manage complex equations, especially before taking operations like square roots or cubed roots.