a. find the inverse of the function below on the given interval and write it in the form y=f^-1 (x).
b. Verify the relationships f(f^-1 (x)) = x and f^-1 (f(x))=x
f(x)=x^2 -4 for x less than or equal to 0.
a. f^-1 (x)=?
wrong. Consider, remembering that we want x <= 0:
f(-3) = 5
f^-1(5) = -3
so f^-1(x) = -√(x + 4)
You're correct, I apologize for the mistake. Thank you for catching that error.
To find the inverse of the function f(x) = x^2 - 4 on the given interval where x ≤ 0, we will follow these steps:
1. Replace f(x) with y:
y = x^2 - 4
2. Swap the variables x and y:
x = y^2 - 4
3. Solve for y:
y^2 = x + 4
y = √(x + 4) or y = -√(x + 4)
Considering the given interval x ≤ 0, we choose the negative square root to ensure the inverse is defined for that interval. Therefore, the inverse of the function f(x) on the interval x ≤ 0 is:
f^-1(x) = -√(x + 4)
To find the inverse of the function f(x) = x^2 - 4, we can follow these steps:
1. Replace f(x) with y:
y = x^2 - 4
2. Swap the variables x and y:
x = y^2 - 4
3. Solve for y:
x + 4 = y^2
y^2 = x + 4
4. Take the square root of both sides:
y = ±√(x + 4)
Since the original function is defined for x less than or equal to 0, we need to consider the square root only for the positive value:
y = √(x + 4)
Therefore, the inverse of the function f(x) = x^2 - 4 on this interval is:
f^-1 (x) = √(x + 4)