Problem 83 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\( f(2) = \sqrt{7} \).

Step 1: Identify the function

The function given is \( f(x) = \sqrt{2x + 3} \). Our task is to evaluate this function at a specific value of \( x \).

Step 2: Substitute the value into the function

We need to compute \( f(2) \). Substitute \( x = 2 \) into the function \( f(x) = \sqrt{2x + 3} \).

Step 3: Calculate the expression inside the square root

Substitute \( 2 \) into \( 2x + 3 \) to get \( 2(2) + 3 = 4 + 3 = 7 \).

Step 4: Compute the square root

Now that we have \( \sqrt{7} \), we calculate it and keep it in this form as it cannot be simplified further to an integer or simpler fraction.

Key Concepts

Square Root Function

Square root functions are a fundamental part of algebraic studies and involve finding the square root of a given expression. The general form of a square root function is written as \( f(x) = \sqrt{g(x)} \), where \( g(x) \) is any function or expression. The square root function graph is characterized by its unique shape as it only exists in the positive domain for real numbers.

• When evaluating a square root function, the expression under the root must always be non-negative for real numbers.
• The function is defined only when the value under the square root is zero or positive, reflecting its graph's nature which starts from a point on the x-axis and moves upwards.

Understanding square root functions is crucial for solving equations involving radicals and simplifies complex algebraic expressions. For example, in the function \( f(x) = \sqrt{2x+3} \), the expression \( 2x+3 \) must be greater than or equal to zero for the function to have real number outputs.

Function Substitution

Function substitution is used to evaluate functions at specific values, allowing us to understand how the function behaves. This technique involves replacing the variable within the function with a given numerical value.

• It essentially converts the variable expression into a constant, helping compute the function's output for that specific input.
• Substitution involves straightforward arithmetic calculations once the value replaces the variable.

To evaluate the function \( f(x) = \sqrt{2x+3} \) at \( x = 2 \), we replace \( x \) with 2, simplifying the expression to \( \sqrt{2(2)+3} = \sqrt{7} \). This process shows the importance of evaluating functions accurately by substituting correctly to gain reliable results.

Algebraic Expression

An algebraic expression involves numbers, variables, and operations that represent a particular value. In algebra, understanding expressions is linked to simplifying and manipulating these, which start with knowing the structure and logic behind them.

• Algebraic expressions can be as simple as \( 4x \) or as complex as \( 2x^2 + 3x + 7 \).
• They form the building blocks for formulating equations and functions.

In the expression \( 2x+3 \), part of the square root function \( f(x) = \sqrt{2x+3} \), simplifying such expressions by performing arithmetic operations is crucial. These steps are essential preceding computing the square root or any further algebraic or arithmetic computation, bridging basic concepts to more complex calculations needed in solving algebra problems.

Radicals in Algebra

Radicals are symbols like square roots that represent the root of a number or expression. They are a key concept in algebra and involve both understanding and manipulating these to make calculations easier.

• The square root symbol, \( \sqrt{} \), is a common type of radical, representing the number which, when multiplied by itself, equals the value under the root.
• It’s essential to know which numbers can and cannot simplify to make computing radicals accurate.

In our function \( f(x) = \sqrt{2x+3} \), when \( x = 2 \), we simplify it to \( \sqrt{7} \), a radical that remains in simplified form since 7 isn't a perfect square. Understanding how to work with and simplify radicals lets you solve and simplify algebra equations effectively, giving clarity and exactness to mathematical solutions.