To find the inverse of the function f(x) = x^12, we need to follow a series of steps. The goal of finding an inverse function is to express the dependent variable (y) in terms of the independent variable (x), and vice versa.
1. Start with the original function:
We begin with:
y = x^12
2. Switch the roles of x and y:
To find the inverse, we swap x and y:
x = y^12
3. Isolate y:
Next, we need to solve for y. To do this, we can apply the twelfth root to both sides of the equation:
y = x^(1/12)
4. Express the inverse function:
We can now write the inverse function in standard notation:
f-1(x) = x^(1/12)
5. Consider the domain:
It’s important to note that the original function f(x) = x^12 is defined for all real numbers; however, it is important to be cautious when interpreting the domain for the inverse function. Since the twelfth root can yield both positive and negative results, the inverse function f-1(x) = x^(1/12) is defined for all real numbers as well.
In summary:
The inverse of the function f(x) = x^12 is f-1(x) = x^(1/12). This means that if you have a number and raise it to the twelfth power, taking the twelfth root of that number will return you to the original number.