Problem 40 - 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. $$ f(x)=\frac{2 x+6}{3} $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{3x - 6}{2} \).

Step 1: Replace f(x) with y

First, replace the function notation \( f(x) \) with \( y \). So, the equation becomes: \( y = \frac{2x + 6}{3} \).

Step 2: Swap x and y

To find the inverse, swap \( x \) and \( y \) in the equation. This gives us \( x = \frac{2y + 6}{3} \).

Step 3: Solve for y

Multiply both sides by 3 to eliminate the fraction: \( 3x = 2y + 6 \). Then, solve for \( y \) by subtracting 6 from both sides: \( 3x - 6 = 2y \). Next, divide everything by 2 to get \( y \): \( y = \frac{3x-6}{2} \).

Step 4: Write the inverse function

Now, express \( y \) in terms of inverse function notation: \( f^{-1}(x) = \frac{3x - 6}{2} \).

Key Concepts

Understanding One-to-One Functions

In the world of mathematics, especially when dealing with functions, the idea of one-to-one functions is crucial. A function is considered "one-to-one" if every y-value has a unique corresponding x-value. Put another way, no y-value is the result of pairing with more than one different x-value.

To visually check if a function is one-to-one, you can use the Horizontal Line Test. If no horizontal line crosses the graph of the function more than once, the function is one-to-one. This property is essential because only one-to-one functions have inverses that are also functions.

• Example: If you have a function that passes the Horizontal Line Test, you can confidently find its inverse and use it in calculations.
• Non-example: A parabola opening upwards or downwards is not one-to-one, as it fails the Horizontal Line Test.

Solving Equations for Inverses

Solving equations is a fundamental skill that allows us to manipulate and rearrange equations to uncover unknown variables or meet specific criteria. In the context of finding an inverse function, solving the equation involves a series of steps.

Firstly, in finding the inverse of a one-to-one function, replace the function notation with y, to simplify the equation. Then, swap the x and y variables. This is because the inverse function essentially reverses the roles of inputs and outputs compared to the original function.

• Example of swapping: For an equation like \( y = \frac{2x + 6}{3} \), switch to \( x = \frac{2y + 6}{3} \).
• Balancing act: Use basic algebra to isolate y, progressively peeling away operations so that y stands alone on one side of the equation.
• Example of solving: To isolate y in \( x = \frac{2y + 6}{3} \), multiply both sides by 3 to clear the fraction, then rearrange to get y by itself.

Using Function Notation

Function notation is a simple and effective way to represent functions and their operations in mathematics. It uses symbols to denote the input and output relationship clearly.

In our exercise, after computing the inverse, we used function notation to denote it as \( f^{-1}(x) \). This expression literally reads as "the inverse of the function f at x."

• Clear and concise: Function notation simplifies complex relations by using symbols like \( f(x) \) or \( g(x) \) instead of cumbersome equations.
• Inverse clarity: Using \( f^{-1}(x) \) clearly communicates that you are talking about an inverse function, a critical distinction when discussing transformations of functions.
• Communicates wide use: This notation isn't just compact; it's universal, consistent across various mathematical studies, making it an invaluable tool as students progress in their learning.