Problem 71 - Elementary and Intermediate Algebra

Question

Let \(f(x)=\frac{1}{x}\) and \(g(x)=\frac{1}{x^{2}} .\) Find each of the following. $$ (g \circ f)(8 x) $$

Step-by-Step Solution

Answer

(g \circ f)(8x) = 64x^2

Step 1: Understanding Composition of Functions

The composition of two functions, like \(g \circ f\), means substituting the function \(f(x)\) into the function \(g(x)\). For the given functions, this means we need to substitute \(f(x) = \frac{1}{x}\) into \(g(x) = \frac{1}{x^2}\).

Step 2: Substitute f(x) into g(x)

Replace \(x\) in \(g(x) = \frac{1}{x^2}\) with \(f(x) = \frac{1}{x}\). This gives us \(g(f(x)) = \frac{1}{(\frac{1}{x})^2}\).

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

Calculate \(g(f(x))\) by simplifying. \[\frac{1}{(\frac{1}{x})^2} = \frac{1}{\frac{1}{x^2}} = x^2.\]Thus, \((g \circ f)(x) = x^2\).

Step 4: Substitute 8x into (g \\circ f)(x)

Now, replace \(x\) with \(8x\) in \(g(f(x)) = x^2\), resulting in \(g(f(8x)) = (8x)^2\).

Step 5: Simplify (8x)^2

Calculate \(g(f(8x))\) by simplifying \((8x)^2\).\[(8x)^2 = 8^2 \cdot x^2 = 64x^2\]So, \(g(f(8x)) = 64x^2\).

Key Concepts

Composition of Functions

The concept of function composition is a method in mathematics where you combine two functions to create a new one. It's like plugging one function into another, which often helps solve complex problems in an easier way. For instance, if you have two functions, say, \( f(x) \) and \( g(x) \), then the composition \( g \circ f \) means that you take \( f(x) \) and use it as an input for \( g \).
This is written as \( g(f(x)) \). Here, we substitute the entire function \( f(x) \) into every place where \( x \) appears in \( g(x) \).
By doing this, we create a single unified expression. Function composition is especially helpful in breaking down problems where direct interactions between different functions are needed.
In the given example, we were asked to find \((g \circ f)(8x)\) for the functions \(f(x)=\frac{1}{x}\) and \(g(x)=\frac{1}{x^{2}}\). This involved substituting one function directly into another.

Simplification of Expressions

Simplifying mathematical expressions is the process of making them easier to work with or understand without changing their value. It is a crucial step that allows mathematicians to work more efficiently and clearly understand relationships within an equation.
In the exercise, after substituting \( f(x) = \frac{1}{x} \) into \( g(x) = \frac{1}{x^2} \), we are left with the expression \( \frac{1}{(\frac{1}{x})^2} \).
We simplify this by:

• Firstly calculating, \( \frac{1}{(\frac{1}{x})^2} = \frac{1}{\frac{1}{x^2}} \).
• This further simplifies to \( x^2 \), because dividing by a fraction is the same as multiplying by its reciprocal.

Simplification helps to transform the complex nested function into a straightforward square of \( x \), \( x^2 \), which is much easier to interpret or use in further calculations.

Substitution Method

The substitution method is a technique used to replace variables or entire expressions with equivalent expressions, making problems easier to solve. It's like swapping out a complicated part of a problem for something simpler or that you've already calculated.
When tackling the problem \( (g \circ f)(8x) \), begin by finding \((g \circ f)(x) = x^2 \). Then, substitute \( 8x \) in place of \( x \) in this expression, following a simple switch.
This results in

• First, substituting it gives \( (8x)^{2} \).
• Then, performing the arithmetic operation to compute \( 8^2 \cdot x^2 = 64x^2 \).

The substitution method is particularly valuable in composition of functions and solving differential equations, as it helps in breaking down larger problems into more manageable steps.