Understanding Inverse Functions
In mathematics, particularly in the realm of algebra and calculus, inverse functions are those functions that reverse the operation of each other. For instance, the inverse of the function g(x) is another function g-1(x) such that g(g-1(x)) x and g-1(g(x)) x. This article will guide you through finding the inverse function of f(x) x^2 - 2x step-by-step.
Steps to Find the Inverse Function of f(x) x^2 - 2x
To find the inverse function of f(x) x^2 - 2x, we will follow these steps:
1. Rewrite the Function
The function f(x) x^2 - 2x can be rewritten in a more manageable form. We complete the square:
f(x) x^2 - 2x (x - 1)^2 - 1This allows us to express the function as: f(x) (x - 1)^2 - 1
2. Set y f(x)
Let y f(x). y (x - 1)^2 - 1
3. Solve for x
We need to solve for x in terms of y. y 1 (x - 1)^2 sqrt(y 1) |x - 1| x - 1 ±sqrt(y 1) x 1 ± sqrt(y 1)
Since f(x) x^2 - 2x is a parabola opening upwards and is not one-to-one over all real numbers, we need to restrict the domain to find a valid inverse. Typically, we choose the right half of the parabola where x ≥ 1.
x - 1 sqrt(y 1) x 1 sqrt(y 1)4. Express the Inverse Function
Hence, the inverse function f-1(y) is:
f-1(y) 1 sqrt(y 1)We can rewrite this as:
f-1(x) 1 sqrt(x 1)Summary: The inverse function of f(x) x^2 - 2x for x ≥ 1 is f-1(x) 1 sqrt(x 1).
Additional Information and Considerations
It is important to note that while solving quadratic equations, the inverse function can have solutions where the base-2 logarithm is necessary for graphical representation, especially when complex values are involved. Additionally, some transformations might be necessary to bring the quadratic equation into sharper relief, as exemplified in the solutions by Mr. Philip Lloyd.
Conclusion
Understanding and finding the inverse function of a given function is crucial for various mathematical operations and applications. By mastering the steps outlined here, you can solve similar problems with confidence.