Problem 39 - Intermediate Algebra

Question

Each of the following functions is one-to-one. Find the inverse of each function and express it using \(f^{-1}(x)\) notation. $$ f(x)=\frac{x-4}{5} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = 5x + 4 \).

Step 1: Write the function equation

Let's start by writing the given function equation: \[ f(x) = \frac{x - 4}{5} \] This is the equation we need to find the inverse for.

Step 2: Replace f(x) with y

Replace \( f(x) \) with \( y \) to set up the equation for finding the inverse:\[ y = \frac{x - 4}{5} \] This will help us rewrite the equation in terms of \( x \) and \( y \).

Step 3: Swap x and y

To find the inverse, swap \( x \) and \( y \) in the equation. This gives:\[ x = \frac{y - 4}{5} \] This equation will be solved for \( y \) to find the inverse function.

Step 4: Solve for y

Multiply both sides of the equation by 5 to eliminate the fraction:\[ 5x = y - 4 \]Add 4 to both sides to isolate \( y \):\[ y = 5x + 4 \] This is the expression for the inverse function.

Step 5: Write the inverse function

Now that we have expressed \( y \) in terms of \( x \), we can write the inverse function:\[ f^{-1}(x) = 5x + 4 \] This is the inverse of the original function.

Key Concepts

One-to-One Functions

One-to-One Functions allow us to find their inverses, which reverses the input-output relationship of the function. A One-to-One Function is a special type of function where each output value is paired with exactly one input value. This property is important in ensuring that the inverse is also a function, as it maintains a unique input-output relationship.

Key characteristics of One-to-One Functions include:

• Each X value (input) produces a unique Y value (output).
• If two outputs from the function are equal, then their inputs must also be equal: if \( f(x_1) = f(x_2) \) then \( x_1 = x_2 \).
• Graphically, they pass the Horizontal Line Test: any horizontal line will intersect the function graph at most once.

Consider the function from the exercise: \( f(x) = \frac{x-4}{5} \). This function is linear and its equation is of the form \( y = mx + b \), which is naturally one-to-one. The linearity guarantees that no horizontal line will intersect the graph more than once, ensuring it is One-to-One and hence, invertible.

Solving Equations

Finding the inverse of a function involves solving equations. By manipulating the function's expression, we reverse the roles of the input and output.

To find the inverse, we can follow these steps:

• Replace the function notation \( f(x) \) with \( y \). This simplifies working with the equation.
• Swap \( x \) and \( y \) to switch the input and output roles. This step sets us up to solve for \( y \).
• Solve the resulting equation for \( y \) to get the inverse. Use algebraic techniques such as adding, subtracting, multiplying, or dividing both sides of the equation to isolate \( y \).

For instance, in the problem \( y = \frac{x - 4}{5} \), swapping \( x \) and \( y \) gives \( x = \frac{y - 4}{5} \). Multiplying the entire equation by 5 eliminates the fraction, yielding \( 5x = y - 4 \). Finally, adding 4 to both sides isolates \( y \), providing the inverse \( f^{-1}(x) = 5x + 4 \). This new expression expresses how to reverse the function's process.

Function Notation

Function Notation is an essential language of mathematics that helps express relationships between variables clearly and concisely. When working with functions, it is important to be comfortable with how they are denoted and manipulated.

Key elements of function notation include:

• Functions are typically denoted by a letter, such as \( f \), \( g \), or \( h \), followed by the variable in parentheses: \( f(x) \).
• The notation \( f(x) = \frac{x-4}{5} \) means that \( f \) is a function in which \( x \) is the variable.
• When functions have inverses, the inverse is usually represented as \( f^{-1}(x) \). This denotes the function that will undo the work done by \( f \).

In this exercise, the original function is given by \( f(x) = \frac{x-4}{5} \). After finding its inverse, we express it using inverse function notation: \( f^{-1}(x) = 5x + 4 \). By mastering function notation, students can better understand how functions operate and communicate their ideas with precision.