Problem 40 - Algebra and Trigonometry

Question

Find the inverse function of \(f\) $$ f(x)=3-5 x $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{x-3}{-5} \).

Step 1: Understand the Function

The given function is \( f(x) = 3 - 5x \). We need to find its inverse function, denoted as \( f^{-1}(x) \). Generally, the inverse function reverses the action of the original function.

Step 2: Replace f(x) with y

To find the inverse, start by replacing \( f(x) \) with \( y \). This gives us the equation \( y = 3 - 5x \).

Step 3: Solve for x

Rearrange the equation \( y = 3 - 5x \) to find \( x \) in terms of \( y \). To do this, subtract 3 from both sides: \( y - 3 = -5x \).

Step 4: Isolate x

Now divide both sides by \(-5\) to isolate \( x \):\[ x = \frac{y - 3}{-5} \]

Step 5: Replace y with x to Express the Inverse Function

The last step is to replace \( y \) with \( x \). Therefore, the inverse function is:\[ f^{-1}(x) = \frac{x - 3}{-5} \]

Key Concepts

Linear Equations

Linear equations are equations of the first degree, meaning they involve no powers higher than one of the variables. A simple form is \( y = mx + b \), representing a straight line when graphed. Here, \( m \) is the slope, indicating the steepness of the line, and \( b \) is the y-intercept, where the line crosses the y-axis.

• Slope (m): Describes the change in \( y \) for a unit change in \( x \). In the function \( f(x) = 3 - 5x \), the slope is -5.
• Y-intercept (b): The value of \( y \) when \( x = 0 \). For \( f(x) = 3 - 5x \), the y-intercept is 3.

Linear equations are fundamental because they represent the simplest function form and are used extensively in various problems, like finding inverse functions as shown in the original problem.

Function Notation

Function notation is a way of representing functions in mathematics by using symbols and is helpful for clarity and simplicity. For example, \( f(x) \) is the traditional notation for a function named \( f \), with \( x \) being its variable or input.
In the equation \( f(x) = 3 - 5x \), \( f \) denotes the function, and \( 3 - 5x \) represents the operation performed on \( x \). This notation helps in distinguishing between different functions and understanding their behavior.

• Input (x): The variable that you substitute into the function.
• Output (f(x)): The result you get after evaluating the function with the input.
• Inverse (f^{-1}(x)): A function that "undoes" the original function, reversing inputs and outputs.

Utilizing function notation, particularly with inverse functions, helps in easily representing and manipulating mathematical relationships.

Solving Equations

Solving equations involves finding the value of variables that make the equation true. It is a critical aspect of algebra that requires several steps and techniques based on the structure of the equation.

• Identifying the Equation: Start by identifying the form of the equation and the unknown variable. For example, when given \( y = 3 - 5x \), \( x \) is the variable you're solving for.
• Rearranging Terms: Use algebraic operations to isolate the variable of interest. Subtraction, addition, multiplication, or division can help move terms across the equation.
• Isolate the Variable: Follow each manipulation step-by-step to solve for the variable. In our example, you subtract 3 and then divide by -5 to extract \( x \).
• Check Your Solution: Substitute the solved values back into the original equation to verify correctness.

Understanding the method of solving equations is essential in deriving the inverse function, a practical application seen in algebra regarding function transformations.