Problem 35 - Algebra and Trigonometry

Question

Find the inverse function of \(f\). \(f(x)=\frac{x}{2}\)

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = 2x \).

Step 1: Replace f(x) with y

The first thing you need to do is replace the function notation \( f(x) \) with \( y \). So, the equation becomes \( y = \frac{x}{2} \).

Step 2: Swap x and y

Now, interchange \( x \) and \( y \). This gives you the equation \( x = \frac{y}{2} \). This step helps us set up the equation for the inverse function.

Step 3: Solve for y

To find the inverse function, solve the equation \( x = \frac{y}{2} \) for \( y \). Multiply both sides by 2 to get \( y = 2x \).

Step 4: Write the inverse function

Replace \( y \) with \( f^{-1}(x) \) to write the inverse function. So, the inverse function is \( f^{-1}(x) = 2x \).

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating these symbols. It allows us to represent mathematical relationships through equations and expressions. When working with inverse functions, algebra helps us manipulate variables to solve for unknowns.
In the exercise, the process of finding an inverse function requires several algebraic steps:

• Replacing variables to shift from an initial equation to a form where the desired variable is isolated.
• Using basic algebraic operations like addition, subtraction, multiplication, and division to simplify and solve equations.

Understanding algebra allows students to rearrange expressions and equations, making it an essential skill when finding inverse functions or handling mathematical tasks in general.

Function Notation

Function notation is a way to represent functions in mathematics clearly and compactly. It uses symbols to express the input and output relationships clearly. For instance, in the given problem, the function is represented as \( f(x) = \frac{x}{2} \). This notation is crucial because it conveys necessary information about the function's behavior.
In function notation:

• \( f(x) \) denotes a function named 'f' with 'x' as its input variable.
• The expression on the right of the equal sign shows how to compute the output from the input.

By learning this notation, you can easily understand and interpret how different inputs affect the output, especially when dealing with more complex functional operations like inverses.

Solving Equations

Solving equations is a core part of algebra and involves finding the values of variables that make the equation true. In the context of inverse functions, solving equations is essential to express one variable in terms of another.
In the exercise above, solving the equation involved swapping variables and isolating the new variable to find the inverse:

• First, the equation \( y = \frac{x}{2} \) was altered by swapping \( x \) and \( y \) to get \( x = \frac{y}{2} \).
• Next, manipulating this equation by multiplying both sides by 2 to isolate \( y \), resulting in \( y = 2x \).

This process shows how equations are manipulated to provide solutions, such as finding the inverse function \( f^{-1}(x) = 2x \), demonstrating algebra's role in removing complexity to solve for variables.