Problem 77 - Intermediate Algebra

Question

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$ f(0) $$

Step-by-Step Solution

Answer

\( f(0) = \sqrt{3} \)

Step 1: Identify the Function for Evaluation

The problem asks us to evaluate the function \( f(x) = \sqrt{2x + 3} \) at \( x = 0 \). This means we need to substitute 0 in place of \( x \) in the given function.

Step 2: Substitute the Value into the Function

Substitute \( x = 0 \) into the function \( f(x) = \sqrt{2x + 3} \). This gives us \( f(0) = \sqrt{2(0) + 3} \).

Step 3: Simplify Inside the Square Root

Simplify the expression inside the square root: \( 2(0) + 3 = 3 \). So, we have \( f(0) = \sqrt{3} \).

Step 4: Find the Final Value of the Function

Since \( f(0) = \sqrt{3} \), this is already in its simplest form. Therefore, \( f(0) = \sqrt{3} \).

Key Concepts

Function Notation

Function notation is a way to represent functions in mathematics. It is a simple, yet powerful tool that allows us to express relationships between variables in a concise manner. For instance, when you see a function represented as \( f(x) \), this indicates that \( f \) is a function dependent on the variable \( x \).
This notation helps us determine the output of a function based on a given input. In our example with \( f(x) = \sqrt{2x + 3} \), \( f(x) \) tells us that for each input \( x \), perform the operations specified to get the result.

• The letter "\( f \)" might change (to \( g, h, etc.\)); it's just a label.
• The variable inside the parentheses (like \( x \)) is the input.
• What you compute by using \( x \) gives you the output.

Square Roots

Understanding square roots is essential when working with certain functions. A square root simply answers the question, "What number, when multiplied by itself, will give a certain value?"
For instance, \( \sqrt{4} \) is 2, because \( 2 \times 2 = 4 \). In equations, square roots are often used to simplify expressions so you can work with them more easily.
In the function \( f(x) = \sqrt{2x + 3} \), the square root is used to find out what influence the expression \( 2x + 3 \) has on the final outcome.

• The symbol "\( \sqrt{} \)" denotes a square root.
• You might also encounter cube roots or higher roots indicated by a little number over the root sign.
• When calculating, the expression inside the square root must first be simplified.

Substitution Method

The substitution method is exactly what it sounds like: substituting a specific value into an equation to evaluate it. This is a crucial step in finding the value of a function for a given input.
In our exercise, we were asked to evaluate \( f(x) = \sqrt{2x + 3} \) at \( x = 0 \). By plugging \( 0 \) into the function in place of \( x \), we get a concrete number that describes the function's value at that specific point.

• The substitution is made by replacing each \( x \) in the function with the given value.
• Ensure accuracy by carefully performing any arithmetic required after substitution.
• This method is widely used because it provides direct answers to questions about specific values of a function.

Algebraic Expressions

Algebraic expressions form the backbone of much of mathematics. They consist of numbers, variables, and operation symbols (like +, −, ⋅, ⁄), and are used to express quantities and relationships. Understanding how to manipulate these expressions is key to solving equations and evaluating functions.
For example, in \( f(x) = \sqrt{2x + 3} \), the part \( 2x + 3 \) is an algebraic expression.

• Algebraic expressions can include constants, coefficients, and variables.
• Operations shown in an expression must be performed in the correct order (following the order of operations rules: parentheses, exponents, multiplication and division, addition and subtraction).
• Expressions can often be simplified by combining like terms or factoring.

By mastering these expressions, you can better understand how functions behave and how to manipulate them to find desired values.