An inverse function reverses the effect of the original function. If we have a function f that maps x to y, then its inverse function f-inverse maps y back to x. This means that f of f-inverse of x equals x, and f-inverse of f of x also equals x. Our goal is to find the inverse of the function f of x equals (x minus 2) divided by (3x plus 1).
To find the inverse of a function, we start by writing y equals f of x. So we have y equals (x minus 2) divided by (3x plus 1). Our goal is to solve this equation for x in terms of y. This will give us the inverse function. First, we'll eliminate the fraction by multiplying both sides by (3x plus 1).
Now we expand the left side of the equation. When we distribute y across the terms in parentheses (3x plus 1), we get 3xy plus y. So our equation becomes 3xy plus y equals x minus 2. We've successfully eliminated the fraction and now have a linear equation in terms of x and y.
Next, we need to collect all terms containing x on one side of the equation and all other terms on the opposite side. Starting with 3xy plus y equals x minus 2, we'll move x to the left side and y to the right side. Subtracting x from both sides gives us 3xy minus x on the left. Subtracting y from both sides gives us negative 2 minus y on the right. So our equation becomes 3xy minus x equals negative 2 minus y.
Now we can factor out x from the left side of the equation. Looking at 3xy minus x, we can factor out x to get x times (3y minus 1). This gives us x times (3y minus 1) equals negative 2 minus y. To solve for x, we divide both sides by (3y minus 1), resulting in x equals (negative 2 minus y) divided by (3y minus 1).
Since we've solved for x in terms of y, we have our inverse function. We can write this as f-inverse of y equals (negative 2 minus y) divided by (3y minus 1). To express this in standard inverse function notation, we replace y with x, giving us f-inverse of x equals (negative 2 minus x) divided by (3x minus 1). Note that there's a domain restriction: since the denominator cannot be zero, x cannot equal one-third. This creates a vertical asymptote in the graph of our inverse function.
To verify our answer is correct, we need to check that f of f-inverse of x equals x. We substitute our inverse function into the original function. This gives us a complex fraction which we can simplify by multiplying numerator and denominator by (3x minus 1). After expanding and combining like terms in both numerator and denominator, we find that they simplify to negative 7x over negative 7, which equals x. This confirms our inverse function is correct.
Let's test our inverse function with specific values to confirm the relationship. First, when x equals 0, f of 0 equals (0 minus 2) divided by (3 times 0 plus 1), which is negative 2 divided by 1, giving us negative 2. Now applying the inverse function to negative 2, we get (negative 2 minus negative 2) divided by (3 times negative 2 minus 1), which is 0 divided by negative 7, giving us 0. As expected, we return to our original value. Let's try another example. When x equals 1, f of 1 equals (1 minus 2) divided by (3 times 1 plus 1), which is negative 1 divided by 4. Applying the inverse function to negative 1/4, we get (negative 2 minus negative 1/4) divided by (3 times negative 1/4 minus 1), which simplifies to negative 7/4 divided by negative 7/4, giving us 1. These examples confirm that our functions truly undo each other!