Problem 72 - Elementary and Intermediate Algebra

Question

Let \(f(x)=\frac{1}{x}\) and \(g(x)=\frac{1}{x^{2}} .\) Find each of the following. $$ (f \circ g)(5 x) $$

Step-by-Step Solution

Answer

\((f \circ g)(5x) = 25x^2\).

Step 1: Understanding the notation

The notation \((f \circ g)(x)\) represents the composition of the functions \( f \) and \( g \). This means that \((f \circ g)(x) = f(g(x))\). In this problem, we need to find \((f \circ g)(5x)\), which implies calculating \(f(g(5x))\).

Step 2: Determine \(g(5x)\)

To begin, substitute \(5x\) into the function \(g(x) = \frac{1}{x^2}\). This gives us: \[ g(5x) = \frac{1}{(5x)^2} = \frac{1}{25x^2}. \]

Step 3: Substitute \(g(5x)\) into \(f(x)\)

Next, substitute \(g(5x)\) into the function \(f(x) = \frac{1}{x}\). This calculates to:\[ f(g(5x)) = f\left( \frac{1}{25x^2} \right) = \frac{1}{\frac{1}{25x^2}}. \]

Step 4: Simplify the expression

Simplify the fraction \(\frac{1}{\frac{1}{25x^2}}\) by taking the reciprocal of the denominator:\[ \frac{1}{\frac{1}{25x^2}} = 25x^2. \]

Step 5: Conclusion

Hence, the composition of the functions \( (f \circ g)(5x) \) is equal to \( 25x^2 \).

Key Concepts

Algebraic Functions

Algebraic functions form a foundational concept in mathematics, often represented as expressions or equations involving variables and constants. These functions can involve operations like addition, subtraction, multiplication, division, and exponentiation.
In the context of function composition, algebraic functions are used to create new functions. For instance, if we have two functions such as \(f(x) = \frac{1}{x}\) and \(g(x) = \frac{1}{x^2}\), they are both algebraic functions. They involve division and exponentiation, which are common operations within this category.

• **Exponents**: Algebraic functions can include terms with exponents, like in \(g(x) = \frac{1}{x^2}\).
• **Fractions**: They can also appear as fractions, like both \(f(x)\) and \(g(x)\) in this case.

Understanding algebraic functions allows you to see how different operations interact with each other, especially when combined through function composition.

Reciprocal Function

A reciprocal function is a special type of function where each non-zero input is mapped to its reciprocal. In simple terms, for a reciprocal function \(f(x)\), you take the number \(x\) and calculate the value \(\frac{1}{x}\). The function \(f(x) = \frac{1}{x}\) is a classic example of this.
For reciprocals:

• **Importance**: They help describe processes where the result diminishes as the input grows larger.
• **Graph**: They usually have a hyperbolic shape, approaching the axes but never touching them.
• **Undefined at Zero**: The function is undefined for zero because division by zero is impossible.

In the exercise, we use reciprocal relationships to simplify and find the value of \(f(g(5x))\). This involves taking the reciprocal of a reciprocal, which effectively simplifies the expression.

Simplifying Expressions

Simplifying mathematical expressions involves reducing them to their simplest form, which makes calculations easier and expressions more understandable. Simplification often requires combining like terms, reducing fractions, and removing unnecessary complexity from expressions.
When simplifying expressions in function composition, such as in \(f(g(5x))\), we follow these general steps:

• **Evaluate Inner Functions First**: Start by finding the inner function's expression, e.g., \(g(5x) = \frac{1}{25x^2}\).
• **Substitute**: Replace the variable in the outer function with this expression, like substituting into \(f\).
• **Simplify**: Look for patterns or mathematical properties that allow further simplification. Here, taking the reciprocal of a reciprocal was the key process.

In our problem, simplifying \(\frac{1}{\frac{1}{25x^2}}\) shows how an initially complex expression can eventually simplify to \(25x^2\), providing clarity and a straightforward answer.