Problem 69 - Elementary and Intermediate Algebra

Question

Let \(f(x)=\frac{1}{x}\) and \(g(x)=\frac{1}{x^{2}} .\) Find each of the following. $$ (g \circ f)\left(\frac{1}{3}\right) $$

Step-by-Step Solution

Answer

\((g \circ f)\left(\frac{1}{3}\right) = \frac{1}{9}.\)

Step 1: Understanding the Composition of Functions

To find \((g \circ f)(x)\), it means to evaluate the composition of functions \(g(f(x))\). In other words, apply \(f\) first and then apply \(g\) to the result.

Step 2: Evaluate Inner Function f(x)

Start with the inner function \(f(x) = \frac{1}{x}\). We need to evaluate \(f\left(\frac{1}{3}\right)\). Substitute \(x = \frac{1}{3}\) into \(f(x)\):\[f\left(\frac{1}{3}\right) = \frac{1}{\left(\frac{1}{3}\right)} = 3.\]

Step 3: Evaluate Outer Function g(f(x))

Now apply the outer function \(g(x)\) using the result from Step 2. Substitute \(x = 3\) into \(g(x) = \frac{1}{x^2}\):\[g\left(f\left(\frac{1}{3}\right)\right) = g(3) = \frac{1}{3^2} = \frac{1}{9}.\]

Step 4: Final Result

The result of the composition \((g \circ f)\left(\frac{1}{3}\right)\) is found to be \(\frac{1}{9}\).

Key Concepts

Inner Function

When dealing with function composition, identifying the **inner function** is crucial. The inner function is the first function applied to the input. In this problem, the function \(f(x) = \frac{1}{x}\) is our inner function. Essentially, the inner function takes the initial input and transforms it into a new value.
Consider the input \(x = \frac{1}{3}\). By inserting this value into our inner function, we transform it as follows:
\[ f\left(\frac{1}{3}\right) = \frac{1}{\left(\frac{1}{3}\right)} = 3. \]

• Start with the given input (in this case, \(\frac{1}{3}\)).
• Plug it into the inner function formula.
• Calculate the result.

The inner function \(f(x)\) simplifies the expression in a way that prepares it for further operations with subsequent functions.

Outer Function

After computing the result of the inner function, we move to the **outer function**. This function acts on the output of the inner function. In our exercise, the outer function is \(g(x) = \frac{1}{x^2}\). The role of the outer function is to further process the result you've obtained from the inner function.
In our current task, the output from the inner function was \(3\). Now, we use this output as the input for the outer function:
\[ g(3) = \frac{1}{3^2} = \frac{1}{9}. \]

• Take the result from the inner function.
• Plug this into the outer function formula.
• Perform necessary calculations to obtain the final result.

The outer function provides the final transformation in the composition, culminating in the desired outcome of \(\frac{1}{9}\).

Evaluate Composition

To **evaluate the composition** of functions, such as \((g \circ f)(x)\), you first identify and calculate each function step-by-step as perspective dictates. Composition essentially layers one function over another, processing in a specific order.
The evaluation follows these steps:
1. Identify your inner and outer functions - here \(f(x)\) and \(g(x)\), respectively. 2. Calculate the inner function using the given input - for example, \(f\left(\frac{1}{3}\right)\). 3. Use the result from the inner function as the input for the outer function. 4. Solve the outer function - such as \(g(f(x)) = \frac{1}{9}\).

• Check each calculation carefully.
• Verify your steps to ensure accurate composition.

The precision and order are essential in function composition to achieve the correct solution, as each layer depends on the preceding results.