Problem 9 - Algebra 2 and Trigonometry

Question

In \(3-10,\) find each of the function values when \(\mathrm{f}(x)=4 x\) $$ \mathrm{f}^{-1}(\mathrm{f}(-6)) $$

Step-by-Step Solution

Answer

The value is \(-6\).

Step 1: Understanding the Task

We are given the function \( f(x) = 4x \), and we need to find the value of \( f^{-1}(f(-6)) \). This involves evaluating the function at \( x = -6 \) and then applying the inverse function.

Step 2: Calculate \( f(-6) \)

To find \( f(-6) \), substitute \( -6 \) into the function: \( f(-6) = 4(-6) = -24 \). So, \( f(-6) = -24 \).

Step 3: Understanding the Inverse Function Concept

The inverse function \( f^{-1} \) is designed to return the original input when you apply \( f \) and then \( f^{-1} \). Thus, \( f^{-1}(f(x)) = x \).

Step 4: Apply the Inverse Function

From the previous step, we know that \( f^{-1}(f(-6)) = -6 \) because applying the inverse function on \( f \) should return the original input of \(-6\).

Key Concepts

Function Evaluation

Function evaluation is an essential step that involves plugging in a specific value into a function to find the corresponding output. For example, we have the function \( f(x) = 4x \). To evaluate this function at \( x = -6 \), we substitute \(-6\) for \( x \). The calculation follows this path: \( f(-6) = 4 \times (-6) \). By performing the multiplication, we get \(-24\). Evaluating functions helps in understanding how inputs are transformed into outputs through algebraic processes. Evaluating the function at different points can reveal the function's behavior and its relationship to other mathematical concepts like inverses. In problems involving multiple steps like inverse functions, clear and correct function evaluation is crucial to ensure accurate results.

Inverse Function Concept

Inverse functions are a crucial concept in understanding how to reverse a function's operation. If \( f(x) \) maps an input \( x \) to an output \( y \), then the inverse function, denoted as \( f^{-1}(x) \), takes \( y \) back to \( x \). The notation \( f^{-1} \) indicates an operation that reverses the work done by the original function.A key property of inverse functions is that applying a function and its inverse in sequence returns you to the original input. Specifically, for any function \( f \), \( f^{-1}(f(x)) = x \). This means if you apply a function to a number and then apply the inverse function to the result, you should end up with the original number. To truly grasp inverse functions, it helps to understand that only bijective (both one-to-one and onto) functions have well-defined inverses. In other words, each element of the output is uniquely mapped from the input, allowing the reversal of the process.

Algebraic Functions

Algebraic functions, like \( f(x) = 4x \), are functions defined using algebraic expressions. They might involve operations such as addition, subtraction, multiplication, and division of variables and constants. Such functions are fundamental in exploring various mathematical concepts, including calculus and inverse functions.The simplicity or complexity of algebraic functions varies. Linear functions like \( 4x \) are straightforward because they form straight lines when graphed. Their behavior is predictable and easy to invert, which is beneficial when dealing with inverse functions.Understanding algebraic functions allows students to figure out how different operations affect movement along the x and y axes. This knowledge is pivotal in solving equations and understanding how changes to the input affect the output. Exploring these operations prepares students for more complex mathematical tasks. In this exercise, algebraic functions beautifully illustrate how a simple function can transform values, which is key when applying inverse operations.