Question

Find (a) \(f \circ g\) and (b) \(g \circ f\). . \(f(x)=\sqrt{x}, \quad g(x)=\sqrt{x}\)

Short answer

The composition of \(f\) and \(g\) (\(f \circ g\)) and the composition of \(g\) and \(f\) (\(g \circ f\)) are both \(x^{1/4}\).

Step-by-step solution

Find \(f \circ g\)

To find the composition of \(f\) and \(g\), denoted as \(f \circ g\), replace every \(x\) in \(f\) with the function \(g(x)\). Thus, \(f \circ g = f(g(x)) = \sqrt{g(x)} = \sqrt{\sqrt{x}}\)

Simplify \(f \circ g\)

Simplify the expression \(\sqrt{\sqrt{x}}\). This can be written as \(x^{1/4}\), since the square root of a square root is the fourth root.

Find \(g \circ f\)

To find the composition of \(g\) and \(f\), denoted as \(g \circ f\), replace every \(x\) in \(g\) with the function \(f(x)\). Thus, \(g \circ f = g(f(x)) = \sqrt{f(x)} = \sqrt{\sqrt{x}}\)

Simplify \(g \circ f\)

Simplify the expression \(\sqrt{\sqrt{x}}\). This can be written as \(x^{1/4}\), since the square root of a square root is the fourth root.

Key concepts

Composite Functions

When studying functions in math, we often encounter situations where we need to apply one function to the result of another function. This process is known as function composition, and it leads to what we refer to as composite functions. To compose two functions, say f(x) and g(x), we apply g(x) first and then apply f(x) to the result, denoted as f(g(x)) or \(f \circ g\).

In the case of our exercise, we have the functions f(x) = \sqrt{x} and g(x) = \sqrt{x}. To find the composition f \circ g, we replace every \(x\) in f with g(x) resulting in \(f \circ g(x) = \sqrt{g(x)} = \sqrt{\sqrt{x}}\). Similarly, we find g \circ f by replacing every \(x\) in g with f(x), leading to \(g \circ f(x) = \sqrt{f(x)} = \sqrt{\sqrt{x}}\). Both composite functions ultimately represent the same expression, which is a testament to the symmetry of this particular combination of functions.

Understanding composite functions is vital for more advanced mathematics, such as calculus, where they come into play in various applications including the Chain Rule, which deals with the derivatives of composite functions.

Radical Expressions

Radical expressions involve roots of numbers or expressions, with the square root being the most common type. The square root of x is written as \(\sqrt{x}\) and represents a value that, when multiplied by itself, gives the original number x. The radical symbol \(\sqrt{}\) itself denotes that we are seeking the root of the quantity inside it.

In the context of our exercise, we are dealing with a nested radical expression: \(\sqrt{\sqrt{x}}\). This might look intimidating at first, but radicals can be approached and simplified systematically. The notion of 'nested' comes from the fact that there is a radical inside another radical. We handle nested radicals by considering the properties of exponents, as roots can be expressed as fractional exponents. For the expression \(\sqrt{\sqrt{x}}\), we are essentially looking for the square root of the square root of x, which can be written as \(\sqrt{x}^{1/2}^{1/2}\) or \(\sqrt{x}^{1/4}\), demonstrating how radical expressions can be simplified and related to exponentiation.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the 'base', to a power, which is the 'exponent'. It tells us how many times to multiply the base by itself. For example, \(x^2\) means \(x\) multiplied by itself, and \(2^3\) means 2 multiplied by itself three times (2 * 2 * 2).

When we deal with radicals, exponentiation comes into play in a slightly different manner. In particular, square roots can be expressed as exponents with a fraction of one-half: \(\sqrt{x}\) is the same as \(x^{1/2}\). Thus, the exercise we are considering simplifies the nested radical expression \(\sqrt{\sqrt{x}}\) to an exponent form by recognizing that the square root of a square root is the fourth root, or \(x^{1/4}\). This representation is often more straightforward to work with, especially in algebra, where manipulating exponents can be more intuitive than dealing with nested radicals. This simplification showcases how exponentiation is not just about whole numbers and can extend into rational numbers, providing a powerful tool for dealing with complicated expressions.