Let f be a function from R to R defined by f(x) = x^2. Find f^–1({x | 0 < x < 1}).
So I found the inverse of f(x):
f^-1(x) = sqrt(x)
But now how do I continue to ensure that it is only for 0 < x < 1?
it is actually good for all x >= 0
The other branch is -√x
You can check the inverse by making sure that (√x)^2 = √(x^2) = x
This is true for all x >= 0
for x < 0, √(x^2) = -x and √x is not real
Thank you for your response. But I would like to know what you would put as a final answer to the problem. Because it seems that the solution I found ( f^-1(x) = √x ) is valid for 0 and ALL positive real numbers because the domain and co-domain are the real numbers. But not all positive real numbers are less than 1, and 0 certainly isn't. So what exactly does "find f^–1({x | 0 < x < 1})" mean?
To ensure that the inverse function f^–1(x) is only for 0 < x < 1, we need to find the corresponding x-values in the original function f(x) within that interval.
To do this, set the inequality 0 < x < 1 and solve for x in terms of y in the original equation f(x) = y.
Given f(x) = x^2 and the inequality 0 < x < 1, rewrite the inequality as 0 < x^2 < 1.
Solving the inequality, we have:
0 < x^2 < 1
Taking the square root of each term, we get:
0 < x < 1
Therefore, for the inverse function to satisfy the condition 0 < x < 1, the corresponding y-values for the inverse function must be in the same range. Thus, the inverse function can be expressed as:
f^–1({x | 0 < x < 1}) = {y | 0 < y < 1}
In other words, the inverse function f^–1({x | 0 < x < 1}) consists of all y-values between 0 and 1.
To find the inverse function of a given function, we first need to express the function in terms of the variable y.
For the function f(x) = x^2, we rewrite it as y = x^2.
To find the inverse function, swap the variables x and y: x = y^2.
Now solve for y:
1. Take the square root of both sides: sqrt(x) = y.
Therefore, the inverse function is f^(-1)(x) = sqrt(x).
Now, to solve for the specific range 0 < x < 1, we need to determine the values of x that satisfy this condition when plugged into the inverse function.
Since x represents the range of values for f(x), we need to check which values of x satisfy the condition 0 < x < 1.
You can do this by substituting values between 0 and 1 into the inverse function f^(-1)(x) = sqrt(x).
For example, if we substitute x = 0.5:
f^(-1)(0.5) = sqrt(0.5) ≈ 0.71
So, the inverse function f^(-1)(x) gives a result between 0 and 1 when 0 < x < 1. By substituting different values of x in this range, you can find all the corresponding values of f^(-1)(x) within that range.