Problem 80 - Algebra 2

Question

Find each composition of functions. Simplify your answer. Let \(f(x)=\frac{1}{x} .\) Find \(f(f(x))\)

Step-by-Step Solution

Answer

The composition of the function \(f(x)=\frac{1}{x}\) with itself is \(f(f(x))=x.\)

Step 1: Understand the function

We have the function \(f(x)=\frac{1}{x}\).

Step 2: Substitute the function into itself

We substitute \(f(x)\) for \(x\) in the given function, which gives us \(f(f(x)) = f(\frac{1}{x})\).

Step 3: Evaluate the function

We now evaluate the function \(f(\frac{1}{x})\). This involves replacing \(x\) with \(\frac{1}{x}\) in the original function, which gives us \(f(f(x)) = \frac{1}{\frac{1}{x}}\)

Step 4: Simplify the function

Simplify the function to get the final answer. A fraction divided by another fraction is equal to the first fraction multiplied by the reciprocal of the second fraction. Hence, \(f(f(x)) = \frac{1}{\frac{1}{x}} = \frac{1}{1} \times x = x.\)

Key Concepts

Functions

Functions are fundamental mathematical objects that map each input to a specific output. Think of a function like a machine; you put something in, it gets processed, and you get something out. The input is known as the domain, and the output is known as the range.

• A function must have only one output for each input. This means if you give a function a specific input, you will always get the same output.
• Functions can be represented in different forms: equations, graphs, tables, or even verbal descriptions.

Understanding how functions work allows us to model and solve real-world problems effectively. Mathematicians often work with more than one function to explore relationships and interconnections between different variables.

Function Notation

Function notation is a concise way of representing functions in mathematics. It looks like this: \(f(x)\). Here, \(f\) denotes the function, and \(x\) is the variable which you are inputting into the function.

• \(f(x)\) can be read as "f of x," and it represents the output of the function when \(x\) is the input.
• In our example, \(f(x) = \frac{1}{x}\) implies that for each value of \(x\), the function gives back its reciprocal as the output.

Function notation is extremely useful because it helps us easily refer to and manipulate functions algebraically. It also provides a clear way to show function composition, like \(f(f(x))\).
By substituting a function into itself or another, we explore how functions combine or transform with each other.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. When it comes to functions, particularly compositions of functions, simplification plays a crucial role.
In our exercise, we simplified \(f(f(x))\) through these steps:

• Substitute: Replace \(x\) with the function's value of it, in this case, \(\frac{1}{x}\).
• Evaluate: Determine \(f\left(\frac{1}{x}\right)\) which is achieved by replacing \(x\) again in \(f(x)\), leading to \(\frac{1}{\left(\frac{1}{x}\right)}\).
• Simplify: Simplify this complex fraction by multiplying by the reciprocal, resulting in \(x\).

Remember, simplification helps in understanding and interpreting mathematical problems more clearly, as well as in their problem-solving application.