Problem 89 - College Algebra

Question

Find the composition when \(f(x)=x^{2}+2\) for all \(x \geq 0\) and \(g(x)=\sqrt{x-2}\). $$ (f \circ g)(11) ;(g \circ f)(11) $$

Step-by-Step Solution

Answer

Both \((f \circ g)(11)\) and \((g \circ f)(11)\) equal 11.

Step 1: Understand the Function Composition

The composition of two functions, \((f\circ g)(x)\), is defined as \(f(g(x))\). Similarly, \((g \circ f)(x)\) is defined as \(g(f(x))\). We have two functions: \(f(x) = x^2 + 2\) and \(g(x) = \sqrt{x - 2}\).

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

Compute \(g(11)\) first for the function \(g\) which is \(\sqrt{x - 2}\). We plug in 11 to get \(\sqrt{11 - 2} = \sqrt{9} = 3\). Next, use this result in function \(f\) which is \(x^2 + 2\). Thus, \(f(3) = 3^2 + 2 = 9 + 2 = 11\). Therefore, \((f \circ g)(11) = 11\).

Step 3: Compute \((g \circ f)(x)\)

Compute \(f(11)\) first for the function \(f\) which is \(x^2 + 2\). We plug in 11 to get \(11^2 + 2 = 121 + 2 = 123\). Next, use this result in function \(g\) which is \(\sqrt{x - 2}\). Thus, \(g(123) = \sqrt{123 - 2} = \sqrt{121} = 11\). Therefore, \((g \circ f)(11) = 11\).

Key Concepts

Algebraic Functions

Algebraic functions are a crucial part of mathematics, often appearing in various forms within equations. These functions are typically constructed using a finite number of algebraic operations such as addition, subtraction, multiplication, division, and taking roots.
Algebraic functions are represented in the form of polynomials, radical expressions, or rational expressions. For example:

• A simple polynomial function could be expressed as: \( f(x) = x^2 + 3x + 5 \).
• A radical function might take the form: \( g(x) = \sqrt{x - 4} \).
• A rational function could appear as: \( h(x) = \frac{x^2 + 2}{x - 3} \).

In solving problems involving algebraic functions, it's essential to understand the domain (the set of all possible input values) of these functions. The domain is determined by the mathematical operations in the function; for example, a square root function like \( \sqrt{x} \) is only defined for non-negative values of \( x \). This domain consideration is important when working with functions like \( g(x) = \sqrt{x-2} \) because it requires \( x - 2 \geq 0 \), leading to \( x \geq 2 \).

Function Notation

Function notation is a systematic way to denote the relationship between input values and their corresponding outputs in a function. When using function notation, we write \( f(x) \) to represent the output of the function \( f \) for the input \( x \).

This notation helps clarify which function is being considered and the variable being used. Function composition, denoted by \((f\circ g)(x)\), is another significant aspect of function notation. It highlights that one function is applied to the result of another function, effectively chaining the operations. Here's how this works:

• \((f \circ g)(x)\) represents \( f(g(x)) \), meaning you first use \( g(x) \) and then apply \( f \) to that result.
• \((g \circ f)(x)\) stands for \( g(f(x)) \), again starting with \( f(x) \) and then applying \( g \) to that output.

This notation simplifies expressing complex operations and allows greater flexibility in solving functions step by step, as seen when calculating \((f \circ g)(11)\) or \((g \circ f)(11)\).

Square Roots

The concept of square roots is fundamental in algebra and beyond. A square root of a number \( a \) is a value \( b \) such that \( b^2 = a \). This implies that the square of \( b \) equals \( a \). In function terms, the square root operation is often represented as \( \sqrt{x} \).

Square roots have specific properties and restrictions:

• For a real-number square root \( \sqrt{x} \), \( x \) must be non-negative (i.e., \( x \geq 0 \)).
• \( \sqrt{x} \) yields a non-negative result.

These aspects are critical when dealing with functions like \( g(x) = \sqrt{x-2} \). For this function, \( x \) must be at least 2, ensuring that the expression inside the square root is non-negative. Calculations involving square roots can often include operations such as simplification or ensuring the operand's eligibility within the function's domain, as demonstrated in the steps to evaluate \((g \circ f)(11)\) where we calculated \( g(123) = \sqrt{121} = 11 \). Understanding these principles is vital for executing successful computations in mathematics.