Question

If \(f\) is a one-to-one function and \(f(3)=8\), then \(f^{-1}(8)=\)_______.

Short answer

3

Step-by-step solution

- Understand the Problem

To find the value of the inverse function at a given point, we need to identify the input of the original function that yields this output.

- Identify Given Information

We are given that the function value is 8 when the input is 3, i.e., \(f(3) = 8\).

- Define the Inverse Function

The inverse function \(f^{-1}(x)\) reverses the roles of inputs and outputs of the original function. This means \(f^{-1}(y) = x\) if \(f(x) = y\).

- Apply Definition to Given Information

Since \(f(3) = 8\), reversing this means that \(f^{-1}(8)\) will give the input for which the output is 8. Thus, \(f^{-1}(8) = 3\).

Key concepts

One-to-One Function

A one-to-one function is a type of function where each input maps to a unique output. This means that no two different inputs produce the same output. For example, if we have a function \(f(x)\), and we know \(f(a) = b\) and \(f(c) = d\), then if \(a eq c\), it must be that \(b eq d\).
This uniqueness is crucial for a function to have an inverse. Without this property, it would be impossible to reliably 'undo' the function and retrieve the original input from an output.

Function Notation

Function notation is a way to represent functions in a standard form. Typically, functions are written as \( f(x) \) where \(f\) is the function name and \(x\) is the input variable. For example, if \(f(x) = 2x + 3\), then \( f(2) = 2(2) + 3 = 7\).
This notation helps to clearly communicate which function is being used and what the input is. Inverse functions are denoted as \( f^{-1}(x) \), signifying that this function reverses the effect of the original function \( f(x) \).

Function Input and Output

Function input and output are the values that you substitute into and get out of a function. If we have a function \(f\), then the input \(x\) is called the 'argument' or 'domain', and the output \(y\) is called the 'value' or 'range'.
For instance, if \(f(x) = x^2\), and you input \(3\), the output will be \(9\). Conversely, for the inverse function \(f^{-1}(y)\), you input the output \(y\) of the original function to get back the initial input \(x\). In our example, since \(f(3) = 8\), the inverse function gives us \(f^{-1}(8) = 3\).

Inverse Function Properties

Inverse functions have key properties that make them unique:

• They reverse the roles of inputs and outputs. If \(f(a) = b\), then \(f^{-1}(b) = a\).
• Combining a function and its inverse yields the original input: \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

Using these properties, we can solve problems effectively. For example, given \(f(3) = 8\), we use the inverse to find that \(f^{-1}(8) = 3\).