Problem 17 - College Algebra

Question

For the following exercises, use each pair of functions to find \(f(g(x))\) and \(g(f(x)) .\) Simplify your answers. $$f(x)=\frac{1}{x-4}, g(x)=\frac{2}{x}+4$$

Step-by-Step Solution

Answer

\(f(g(x)) = \frac{x}{2}\) and \(g(f(x)) = 2x - 4\).

Step 1: Find \(f(g(x))\)

To find \(f(g(x))\), substitute the expression for \(g(x)\) into the function \(f(x)\). Therefore, replace \(x\) in \(f(x) = \frac{1}{x-4}\) with \(g(x) = \frac{2}{x}+4\), resulting in:\[f(g(x)) = \frac{1}{\left(\frac{2}{x} + 4\right) - 4}\]

Step 2: Simplify \(f(g(x))\)

Simplify the expression obtained in Step 1. First, simplify the denominator:\[\frac{2}{x} + 4 - 4 = \frac{2}{x}\]So, the function becomes:\[f(g(x)) = \frac{1}{\frac{2}{x}}\]Simplify by multiplying by the reciprocal:\[f(g(x)) = \frac{x}{2}\]

Step 3: Find \(g(f(x))\)

To find \(g(f(x))\), substitute the expression for \(f(x)\) into the function \(g(x)\). Therefore, replace \(x\) in \(g(x) = \frac{2}{x} + 4\) with \(f(x) = \frac{1}{x-4}\), resulting in:\[g(f(x)) = \frac{2}{\left(\frac{1}{x-4}\right)} + 4\]

Step 4: Simplify \(g(f(x))\)

Simplify the expression obtained in Step 3. First, simplify the fraction:\[\frac{2}{\frac{1}{x-4}} = 2(x-4)\]Thus, substitute back into the equation:\[g(f(x)) = 2(x-4) + 4\]Simplify this expression:\[g(f(x)) = 2x - 8 + 4 = 2x - 4\]

Key Concepts

functions in algebra

In algebra, functions are mathematical entities that establish a relationship between two variables. These variables are typically represented as x and y. For every value of x, a function produces exactly one value of y. This idea is a cornerstone of algebra, enabling us to model and solve real-world and theoretical problems.

When dealing with functions in algebra, you'll often encounter different types, including linear, quadratic, polynomial, and rational functions. Understanding how these functions operate and interrelate is essential.

• **Function Composition**: This involves taking two functions and creating a new function. If you have two functions, say, f(x) and g(x), their composition is denoted as f(g(x)) or g(f(x)). This means you replace the x in one function with the other function.
• **Evaluating Functions**: When evaluating, say, f(g(x)), you substitute g(x) into f. This process requires precise attention to both the operational rules of algebra and arithmetic manipulations.

Understanding these principles of function composition is critical to making sense of how complex algebraic structures interact.

rational functions

Rational functions are a specific type of function expressed as the ratio of two polynomials. In other words, a rational function has the form \( f(x) = \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials. These functions are particularly interesting because they can exhibit behavior such as asymptotes and discontinuities, where the function is not defined.

Key aspects of rational functions include:

• **Domain Restrictions**: Since division by zero is undefined, the values of x for which the denominator \( q(x) = 0 \) are excluded from the domain of the function. Therefore, identifying these values is a crucial part of working with rational functions.
• **Simplification**: Simplifying rational functions is often necessary to understand their behavior more clearly. This might involve factoring polynomials, cancelling like terms, or finding the least common denominator to unify expressions.

In exercises involving rational functions, like our original problem, being adept at substituting expressions and simplifying them is key to finding the correct composition of functions.

algebraic simplification

In algebra, simplification is the process of transforming an expression into a simpler or more usable form. This often involves reducing fractions, combining like terms, or clearing out complex denominators. Algebraic simplification is vital because it allows us to understand, compare, and work with equations or functions more easily.

Consider these steps:

• **Breaking Down Complex Expressions**: When presented with a compound function, break it down into manageable parts. Simplify the denominator first in rational functions, as seen in \( f(g(x)) \).
• **Using Reciprocal Multiplication**: Multiplying by the reciprocal is a common tactic to simplify fractions, as shown when simplifying \( f(g(x)) = \frac{1}{\frac{2}{x}} \) to \( \frac{x}{2} \).
• **Combining Like Terms**: When simplifying expressions like \( g(f(x)) = 2(x-4) + 4 \), distribute and combine the terms to get a final simplified version, \( 2x - 4 \).

Understanding these algebraic simplification techniques will not only help in solving assignments but also enhance overall problem-solving skills in mathematics.