Problem 31 - Intermediate Algebra

Question

Determine the indicated functional values. (Objective 2 ) If \(f(x)=\frac{1}{x}\) and \(g(x)=\frac{2}{x-1}\), find \((f \circ g)(2)\) and \((g \circ f)(-1)\).

Step-by-Step Solution

Answer

\((f \circ g)(2) = \frac{1}{2}\) and \((g \circ f)(-1) = -1\).

Step 1: Understand Function Composition

The composition of two functions \(f\) and \(g\), denoted as \((f \circ g)(x)\), means you apply \(g\) first, and then apply \(f\) to the result of \(g(x)\). Similarly, \((g \circ f)(x)\) means applying \(f\) first, followed by \(g\).

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

First, compute \(g(2)\). The function \(g(x) = \frac{2}{x-1}\). Substitute \(x = 2\) into \(g\): \[g(2) = \frac{2}{2-1} = \frac{2}{1} = 2.\] Now, apply \(f\) to this result: \(f(x) = \frac{1}{x}\). Thus, \[f(g(2)) = f(2) = \frac{1}{2}.\]

Step 3: Compute \((g \circ f)(-1)\)

First, compute \(f(-1)\). The function \(f(x) = \frac{1}{x}\). Substitute \(x = -1\) into \(f\): \[f(-1) = \frac{1}{-1} = -1.\] Next, apply \(g\) to this result: \(g(x) = \frac{2}{x-1}\). Substitute \(-1\): \[g(f(-1)) = g(-1) = \frac{2}{-1-1} = \frac{2}{-2} = -1.\]

Key Concepts

Functional Values

Understanding functional values is crucial for tackling problems related to function composition. In mathematics, a function takes an input value, performs some kind of operation or transformation, and outputs a new value. Functional values refer specifically to these outputs that a function gives when you substitute different inputs into it.
For example, for the function \(f(x) = \frac{1}{x}\), if you input 2, the functional value is \(\frac{1}{2}\). Similarly, for \(g(x)=\frac{2}{x-1}\) and input of 2 yields the functional value 2. By understanding these outputs, students can better navigate through operations like function composition and solve related problems more confidently.

Algebraic Functions

Algebraic functions are a type of mathematical expression built from polynomials, roots, and rational expressions. They form the foundation of many equations and functions you encounter in algebra and calculus.
In our example problem, both \(f(x) = \frac{1}{x}\) and \(g(x)=\frac{2}{x-1}\) represent rational algebraic functions. These algebraic functions are utilized to manipulate inputs and derive specific outcomes or functional values. Recognizing and working with algebraic functions enhance one's ability to explore not only standalone operations but also more complex operations like compositions and inverses.

Step-by-step Solution

What makes solving a math problem manageable is breaking it into smaller and more understandable steps. Let's take the composition of functions as explored in the solution for some clarity:

• First, understand the problem: What operations are you being asked to perform?
• Identify the order of function application: Apply \(g(x)\) before \(f(x)\) for \((f \circ g)(x)\) and the reverse for \((g \circ f)(x)\).
• Always compute the inner function first. For instance, to find \((f \circ g)(2)\), compute \(g(2)\) first, then find \(f(g(2))\).
• Finally, do the operations methodically, substituting values into the equations step-by-step.

This structured approach helps break down potentially complex operations into manageable tasks, making the problem less intimidating and improving problem-solving skills.