Problem 12 - Algebra A Combined Function

Question

Given the one-to-one function \(f(x)=x^{3}+2,\) find the following. (Hint: You do not need to find the equation for \(f^{-1}\).) a. \(f(0)\) b. \(f^{-1}(2)\)

Step-by-Step Solution

Answer

a) \(f(0) = 2\); b) \(f^{-1}(2) = 0\).

Step 1: Evaluate f(x) at x = 0

Start by substituting 0 into the function \(f(x) = x^3 + 2\). This means calculating \(f(0)\) which gives us \((0)^3 + 2 = 2\). Thus, \(f(0) = 2\).

Step 2: Understanding the Inverse Function

The inverse function \(f^{-1}(x)\) reverses the operation of the original function \(f(x)\). Since \(f(x)\) outputs the value \(2\) when \(x = 0\), the inverse function \(f^{-1}(x)\) will output \(0\) when \(x = 2\). So, \(f^{-1}(2) = 0\).

Key Concepts

one-to-one functions

A one-to-one function is a special kind of function where each input has a unique output, and no outputs are the same for different inputs. This property is crucial in determining inverse functions, as for a function to have an inverse, it needs to be one-to-one.

Think of a one-to-one function as a strict rule-following machine. Each input gets mapped to only one output, and that output isn't shared with any other inputs. This makes it possible to "reverse" the function, or in other words, find an inverse function that undoes the action of the original.

• Example: In the function given, \(f(x) = x^3 + 2\), if you input any number, you get a unique result.
• Importance: Identifying one-to-one functions helps us to properly find inverse functions.
• Visualizing: Imagine a line that doesn't touch any horizontal line more than once on a graph—such a line represents a one-to-one function.

Understanding one-to-one functions is vital because it ensures when we talk about inverse functions, they truly exist and are properly defined.

evaluating functions

Evaluating functions involves finding the output of the function for a specific input. This process is straightforward when following the function's formula. Let's see how we do this step-by-step with the function \(f(x) = x^3 + 2\).

First, substitute the given value into the equation. For example, when evaluating \(f(0)\):

• Step 1: Replace every \(x\) in the equation with \(0\).
• Step 2: Simplify \((0)^3 + 2 = 0 + 2 = 2\).
• Result: So, \(f(0) = 2\).

This process is about substituting and simplifying, which makes it very logical and easy to handle.

Additional tips:

• Always be careful with signs and arithmetic operations.
• Double-check your calculations to avoid errors.
• Practice with different values to improve your confidence and understanding.

function notation

Function notation is a way to denote and work with functions in math. It tells us which function to use and the input value at the same time. The notation \(f(x)\) denotes a function named \(f\) with \(x\) as the input. It's very useful for organizing and solving problems systematically.

Here's how to navigate function notation effectively:

• \(f(x)\) means apply the function to \(x\).
• Notations like \(f(0)\), \(f(1)\) tell you precisely what to plug into the function to evaluate it.
• Leaving the input as a variable, such as \(f(x)\), allows you to see how the function behaves with any number.

This notation is not limited to lowercase letters—a function can be denoted with any letter or even a symbol. Importantly, the notation \(f^{-1}(x)\) represents the inverse function, which reverses what \(f(x)\) does.

Understanding function notation is like having a toolbox, where each tool (notation) has a specific use case, ensuring your calculations and problem-solving are organized and clear.