Problem 38 - Algebra and Trigonometry

Question

Find the inverse function of \(f\) $$ f(x)=6-x $$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = 6-x\).

Step 1: Replace f(x) with y

We start by replacing the function notation \(f(x)\) with \(y\). This gives us the equation \(y = 6-x\).

Step 2: Swap x and y

To find the inverse function, we interchange the roles of \(x\) and \(y\). This means we now write the equation as \(x = 6-y\).

Step 3: Solve for y

Next, we solve for \(y\) by isolating it on one side of the equation. Start with \(x = 6-y\), add \(y\) to both sides to get \(x + y = 6\), and then subtract \(x\) from both sides to get \(y = 6-x\).

Step 4: Express the inverse function

Since \(y\) is expressed in terms of \(x\) as \(y = 6-x\), the inverse function is \(f^{-1}(x) = 6-x\).

Key Concepts

Function Notation

Function notation is a simple yet essential way of writing mathematical functions, usually denoted as \( f(x) \). Here, "\( f \)" represents the function, and "\( x \)" represents the input value for which the function \( f \) calculates an output. Think of it as a machine where an input goes in, and an output comes out.

• \( f(x) \) implies that "\( f \)" depends on "\( x \)".
• Instead of writing long expressions repeatedly, function notation helps simplify and condense such expressions.
• It also makes it easy to represent inverse functions and explore their characteristics.

In our problem, \( f(x) = 6 - x \) simply means, for any input \( x \), the output will be calculated as \( 6 - x \). Function notation allows us to seamlessly transition between original functions and their inverses by just adjusting the input and output roles between \( x \) and \( y \).

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. Think of it as understanding how the inputs relate to the outputs by reversing their roles if necessary, especially when we want to find inverse functions.

• An equation like \( y = 6 - x \) means that we want to determine what "\( y \)" equals for any given \( x \).
• To find an inverse function, you often have to rearrange an equation to isolate the variable of interest.

In this particular exercise, the goal was to find the expression for \( y \) when given \( x = 6 - y \). This required adding \( y \) to both sides and simplifying to solve for \( y \). These steps are crucial as they allow you to see the relationship between the variables from a different perspective, ultimately leading to understanding how changes in one variable affect the other.

Interchanging Roles of Variables

Finding the inverse function requires us to interchange the roles of the dependent variable, usually "\( y \)," and the independent variable, usually "\( x \)." This means we swap what is considered input and output. It's a bit like solving a puzzle from the opposite direction.

• Start with the original equation, \( y = f(x) \).
• Interchange the variables to see \( x \) now as \( y \) and \( y \) as \( x \).
• This reformulation is crucial; it allows us to find the inverse equation, showing how to convert outputs back into inputs.

For the exercise, swapping the \( x \) and \( y \) in \( y = 6 - x \) to \( x = 6 - y \) was a key step. This change enables solving for the new "\( y \)," which provides the inverse function \( f^{-1}(x) \). This swapping is a powerful technique to uncover and explore the deeper connections between variables in mathematical relationships.