Finding the Inverse of a One-to-One Function: fx 1 / (-3x - 1)
Understanding the concept of an inverse function is crucial in mathematics, especially in the field of algebra. In this article, we will walk through the process of finding the inverse of a one-to-one function, specifically the function given by fx 1 / (-3x - 1). Understanding this process will not only enhance your mathematical skills but also prepare you for more complex problems in calculus and beyond.
Introduction to Inverse Functions
In mathematics, an inverse function is a function that "reverses" another function. More specifically, if y f(x) is a one-to-one function, then x f-1(y) is its inverse function. To find the inverse of a function, we follow a set of steps that involve algebraic manipulation. In this article, we will apply these steps to the specific function fx 1 / (-3x - 1).
Step 1: Swap Variables
The first step in finding the inverse of a function is to switch the x and y variables. This means we start with the given function:
fx 1 / (-3x - 1)
Swap x and y:
y 1 / (-3x - 1)
Step 2: Solve for y
Next, we solve the equation for y. This is the core step in the process of finding the inverse function.
Start by isolating y on one side of the equation: Multiply both sides by the denominator to get rid of the fraction: Expand and simplify the equation: Rearrange the terms to solve for y:Oftentimes, this step involves algebraic manipulation to isolate y. Let's follow these steps with the given function:
Start with the swapped equation: Multiply both sides by the denominator: Expand and simplify: Rearrange the terms:So, we have:
x(-3y - 1) -1
x(-3y - 1) -1
x(-3y - 1) -1 (Expand)
-3xy - x -1 (Distribute x)
-3xy x - 1 (Isolate the y term)
y (x - 1) / (-3x) (Divide by -3x to isolate y)
Therefore, the inverse function is:
f-1(x) (x - 1) / (-3x)
Step 3: Simplify the Inverse Function
While the inverse function is already in a simplified form, we can further simplify it for clarity. After some algebraic manipulation, we get:
f-1(x) 1 - x / 3x
Conclusion
In conclusion, finding the inverse of a one-to-one function is a valuable skill in mathematics. By following the steps of swapping variables and solving for y, we can find the inverse of the given function f(x) 1 / (-3x - 1) and arrive at the final result, which is f-1(x) 1 - x / 3x. Understanding this process is not only important in algebra but also in more advanced mathematical fields such as calculus and differential equations.