Question

\(f(x)=4-x^2\)

Short answer

The roots of the function \(f(x) = 4 - x^2\) are \(x = 2\) and \(x = -2\).

Step-by-step solution

Rearrange the equation

First, rearrange the equation from \(4 - x^2 = 0\) to \(x^2 = 4\), by moving \(x^2\) to one side and \(4\) to the other.

Apply square root

Next, apply square root on both sides of the equation: \(x = ± sqrt(4)\).

Simplify the equation

On simplifying, \(x = ±2\) which means the function has two roots, at \(x = 2\) and \(x = -2\).

Key concepts

Understanding Algebraic Functions

When tackling algebra, one fundamental concept we encounter is that of algebraic functions. An algebraic function, such as the one in the exercise, \(f(x) = 4 - x^2\), is a type of function that can be written using polynomial expressions. They are essentially rules that take an input (in our case, the variable \(x\)) and produce an output after performing a series of algebraic operations.

In the context of our exercise, the function \(f(x)\) takes a value of \(x\) and squares it (\(x^2\)), and then subtracts this value from 4. The beauty of algebraic functions lies in their predictability; for every input, there is a definite output. This characteristic allows us to solve for \(x\) by setting the output to zero and finding what inputs (or \(x\) values) make the equation true, which leads us to the notion of roots of a function.

Demystifying the Square Root

The square root is a mathematical operation that can be thought of as the opposite of squaring a number. If squaring a number means multiplying the number by itself (for example, \(3^2 = 9\)), then finding the square root of a number means discovering what original number, when squared, would produce the given number. It's denoted as \(\sqrt{x}\).

For instance, in our step-by-step solution, finding the square root of 4 is required to solve the equation. The square root of 4 is 2, because when 2 is squared (\(2^2\)), the result is 4. However, there's a unique twist to square roots: each positive number has two square roots, a positive and a negative. This is why from \(x^2 = 4\), we get two solutions: \(x = \pm2\). Thus, understanding square roots is essential when solving quadratic equations, as it often leads us to the possible values of \(x\) that satisfy the equation.

Finding the Roots of a Function

Roots of a function, in algebra, are the values for which the function equals zero. These are critical as they often represent points of interest in various applications, such as physics and engineering. In the given problem, \(f(x) = 4 - x^2\), finding the roots involves determining the values of \(x\) that make \(f(x)\) equal to zero.

Following the steps laid out in the exercise, after rearranging and applying the square root, we arrive at two solutions, \(x = 2\) and \(x = -2\), indicating that our function intersects the \(x\)-axis at these points. In a graphical perspective, these would be the points where the curve of the function touches or crosses the \(x\)-axis. Recognizing these roots is not only pivotal in solving equations but also in graphing and interpreting the behavior of algebraic functions.