Problem 19 - Brief Applied Calculus

Question

For each function: $$ f(x)=\sqrt{4-x^{2}} ; \text { find } f(0) $$

Step-by-Step Solution

Answer

f(0) = 2.

Step 1: Identify the Function and Value to Substitute

The function given is \( f(x)=\sqrt{4-x^2} \). You need to find the value of this function when \( x = 0 \).

Step 2: Substitute the Value of x

Substitute \( x = 0 \) in the function: \( f(0) = \sqrt{4-0^2} \).

Step 3: Simplify the Expression

Simplify the expression inside the square root: \( 4 - 0^2 = 4 \). This simplifies to \( f(0) = \sqrt{4} \).

Step 4: Calculate the Square Root

Calculate \( \sqrt{4} \), which equals 2. Therefore, \( f(0) = 2 \).

Key Concepts

Understanding Square Roots

Square roots are fundamental in mathematics. When we talk about the square root of a number, we're looking for a value which, when multiplied by itself, gives the original number. For example, the square root of 4 is 2 because when 2 is multiplied by 2, you get 4.
Considered a principal operation, we often use the symbol \( \sqrt{} \) to indicate the square root. This operation reverses the effect of squaring a number, providing a way to solve quadratic equations.
Here are a few key points about square roots:

• Every positive number has two square roots: one positive (the principal square root) and one negative.
• Square root of zero is zero.
• Negative numbers do not have real square roots because no real number squared gives a negative product.

In solving mathematical problems, recognizing and computing square roots can be crucial in simplifying expressions and solving equations. In our exercise, calculating \( \sqrt{4} \) gives us a decisive step towards finding our function's value.

The Process of Function Evaluation

Function evaluation involves determining the output of a function given a specific input. In mathematical terms, if we have a function \( f(x) \), evaluating it at a point means substituting that point into the function. For example, in our exercise, we are asked to find \( f(0) \) for the function \( f(x) = \sqrt{4 - x^2} \).
To successfully evaluate a function, follow these simple steps:

• Identify the given function and the input value you need to evaluate it at.
• Substitute the input value into the function, replacing the variable (typically \( x \)).
• Simplify the resulting expression to find the function's value at that point.

This approach allows us to verify what specific output a function gives at particular values, paving the way to understanding its behavior and any potential real-world implications.

Exploring the Substitution Method

The substitution method is a handy technique often used in algebra, calculus, and other areas of mathematics. It allows us to replace a variable with a specific value or expression to simplify and solve equations.
Let's break down how substitution works using our example:

• We start with a function: \( f(x) = \sqrt{4 - x^2} \) and a task to evaluate it at \( x = 0 \).
• We substitute 0 for \( x \) in the function: \( f(0) = \sqrt{4 - 0^2} \).
• Next, we simplify the expression inside the square root \( \sqrt{4} \), eventually calculating to find that \( f(0) = 2 \).

This method, widely applicable, simplifies finding precise values of functions, solving equations, or integrating complex expressions by reducing them to simpler forms. It is a foundational tool in mathematics, making seemingly complicated problems more manageable.