Problem 62 - Precalculus Essentials

Question

Find. a. \((f \circ g)(x)\) b. \((g \circ f)(x)\) c. \((f \circ g)(2)\) d. \((g \circ f)(2)\) $$f(x)=\sqrt{x}, g(x)=x+2$$

Step-by-Step Solution

Answer

\((f \circ g)(x)=\sqrt{x+2}\), \((g \circ f)(x)=\sqrt{x}+2\), \((f \circ g)(2)=2\), \((g \circ f)(2)=\sqrt{2}+2\).

Step 1: Calculating \((f \circ g)(x)\)

Replace \(x\) in \(f(x)=\sqrt{x}\) with \(g(x)=x+2\). So \(f(g(x))=\sqrt{x+2}\).

Step 2: Calculating \((g \circ f)(x)\)

Replace \(x\) in \(g(x)=x+2\) with \(f(x)=\sqrt{x}\). So \(g(f(x))=\sqrt{x}+2\).

Step 3: Calculating \((f \circ g)(2)\)

Substitute \(2\) into the composite function \(f(g(x))=\sqrt{x+2}\) to give \(f(g(2))=\sqrt{2+2}=\sqrt{4}=2\).

Step 4: Calculating \((g \circ f)(2)\)

Substitute \(2\) into the composite function \(g(f(x))=\sqrt{x}+2\) to give \(g(f(2))=\sqrt{2}+2\).

Key Concepts

Function Composition

Function composition is a fundamental concept in precalculus that deals with combining two or more functions to form a new function. This involves taking the output from one function and using it as the input for another function. The notation \(f \text{ and } g\) are functions, then their composition is denoted by \(f \text{ and } g(x)\), which is read as 'f of g of x'. In our exercise, we have the functions \(f(x) = \sqrt{x}\) and \(g(x) = x + 2\). When we calculate the composition \(f \text{ and } g(x)\), we replace the \(x\) in \(f\) with \(g(x)\), resulting in a new function \(f(g(x)) = \sqrt{x+2}\).

Conversely, for \(g \text{ and } f(x)\), we replace the \(x\) in \(g\) with \(f(x)\), yielding the function \(g(f(x)) = \sqrt{x} + 2\). This operation creates a hierarchy of functions, where one function's output feeds directly into the next, thus chaining them together. It requires careful attention to the order of operations, as reversing the functions generally yields a different result.

Precalculus

Precalculus serves as the foundation to many mathematical concepts that are expanded upon in calculus. It usually encompasses the study of functions, their properties, and operations like function composition. Knowledge of precalculus is essential for understanding and solving the exercise provided. Here, the student must not only be familiar with basic function operations but also be adept in manipulating functions to find composite functions.

In the context of our problem, precalculus involves understanding how to work with square root functions and how to compose them with linear functions. The ability to do so is an important skill set that builds the groundwork for more advanced courses in mathematics, such as calculus, where the idea of 'function of function' appears frequently in concepts like the chain rule.

Square Root Functions

Square root functions are a type of radical function and can be represented as \(f(x) = \sqrt{x}\). These functions are significant in precalculus and give us the principal square root of the input value \(x\). In our exercise, the function \(f(x) = \sqrt{x}\) is a square root function. When dealing with these, it's important to remember that they are defined only for \(x \geq 0\), as square roots of negative numbers are not real numbers in the standard real number system.

The graph of a square root function is a curve that increases slowly and never goes below the \(x\)-axis, emphasizing that it only represents non-negative outputs. In function composition, when a square root function is involved, careful consideration must be given to the range of the other function to ensure that its outputs will result in real number outputs after taking the square root.

Function Operations

Function operations include addition, subtraction, multiplication, division, and composition of functions. These operations allow us to build new functions from existing ones. In the textbook problem we're looking at, we're primarily concerned with function composition, which combines two functions in a specific order. Nonetheless, understanding all operations is key to mastering precalculus.

The steps provided in the solution demonstrate the composition operation, one of the more complex function operations due to its nesting nature. Each function operates on the other, and care must be taken to evaluate them in the correct sequence. This distinction in the order is paramount, as \(f \text{ and } g(x)\) and \(g \text{ and } f(x)\) are generally not the same. Practicing these operations helps build intuition for how functions interact and can pave the way for solving more intricate problems that involve multiple layers of function operations.