Problem 107 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$\left(f^{-1} \circ f^{-1}\right)(-6)$$

Step-by-Step Solution

Answer

The value of \( \left(f^{-1} \circ f^{-1}\right)(-6) \) is -168.

Step 1: Understanding the Problem

We're given functions \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\). We are asked to find the value of \( \left(f^{-1} \circ f^{-1}\right)(-6) \). This means applying the inverse function of \( f \) twice on the input -6.

Step 2: Find the Inverse of Function f

The first struggle would be to find the inverse function. To find the inverse function \( f^{-1} \), we first swap \( x \) and \( y \) in the equation and then solve for \( y \). Our starting function is \( f(x)=\frac{1}{8} x-3 \), so we substitute \( x \) and \( y \) to get \( x=\frac{1}{8} y-3 \). Solving this for \( y \) we get \( y=8x+24 \). This is our \( f^{-1}(x) \).

Step 3: Applying the Inverse Function

So now we apply \( f^{-1} \) twice on -6. This means we first substitute \( x=-6 \) into \( f^{-1}(x) \) and then substitute the result again into \( f^{-1}(x) \). Calculating \( f^{-1}(-6)=8*(-6)+24=-24 \). Calculating \( f^{-1}(-24)=8*(-24)+24=-168 \).

Step 4: Final Result

Therefore, \( \left(f^{-1} \circ f^{-1}\right)(-6)=-168 \).

Key Concepts

Function Notation

Function notation is a way to denote functions in mathematics. It allows you to clearly express how a function is supposed to operate on an element. This notation is very helpful for identifying which function you are working with and the exact inputs and outputs. It is expressed as follows:

• A function named "f" applied to an input "x" is written as \( f(x) \). The value where "x" is the argument of the function.
• If you have two functions, such as \( f(x) \) and \( g(x) \), and you want to compose them or combine them in a series of operations, you use that same notation.
• Furthermore, the notation \( f^{-1}(x) \) signifies the inverse of the function \( f \).

Understanding this notation is critical as it allows you to follow calculations and identify processes accurately.

Composition of Functions

The composition of functions involves combining two or more functions to produce a new function. This is represented by \((f \circ g)(x)\), which means you first apply the function \(g\) to the input \(x\), and then apply \(f\) to the result of the first application. In simpler terms, it is like a chain reaction:

• Start with input \(x\).
• Apply the first function \(g\) to this input.
• Take the output from \(g(x)\) and apply the second function \(f\) to it.

When dealing with inverse functions, as in the original exercise, the process involves applying the inverse function twice. The notation \( (f^{-1} \circ f^{-1})(x) \) illustrates how a function can be repeatedly applied, using its inverse, to achieve a desired outcome.

Algebraic Manipulation

Algebraic manipulation involves using math operations to modify expressions in a desired way. This often includes steps like isolating terms, expanding equations, or simplifying expressions. Here's how algebraic manipulation was handled in the inverse functions example:

• To find \( f^{-1} \), you swapped \(x\) and \(y\) in the original function and then solved for \(y\). This was done by first writing the equation \(x = \frac{1}{8} y - 3\).
• The next step was solving for \(y\), which involved undoing operations one at a time. For example, adding 3 to both sides to deal with subtraction, and multiplying by 8 to undo division.
• Each step requires simplifying the equation until it is in the form \(y = \) expression that contains only one variable.

Algebraic manipulation is foundational, as it is applied to find inverse functions, solve equations, and simplify mathematical expressions. Knowing how to manipulate algebraically gives you more tools to work with every kind of equation.