what is the inverse function of f(x)=-[sqrt(25-x^2]
It might look like it'd be
y = sqrt(25 - x^2)
The problem is that each quarter-circle has its own inverse.
There is no inverse function for the lower half-circle, since each y-value has two x-values. When trying to invert f(x), there is no way to decide which to pick.
To find the inverse function of f(x) = -√(25 - x^2), we can follow the steps below:
Step 1: Replace f(x) with y.
y = -√(25 - x^2)
Step 2: Swap x and y.
x = -√(25 - y^2)
Step 3: Solve for y.
x = -√(25 - y^2)
Step 4: Square both sides to eliminate the square root.
x^2 = 25 - y^2
Step 5: Rearrange the equation to solve for y^2.
y^2 = 25 - x^2
Step 6: Take the square root of both sides.
y = ±√(25 - x^2)
Since y can be both positive and negative, we have two possible inverse functions:
f^(-1)(x) = √(25 - x^2)
f^(-1)(x) = -√(25 - x^2)
Therefore, the inverse function of f(x) = -√(25 - x^2) is f^(-1)(x) = √(25 - x^2) and f^(-1)(x) = -√(25 - x^2).
To find the inverse function of f(x) = -√(25-x^2), we need to swap the roles of x and f(x) and solve for x.
Step 1: Let's start by replacing f(x) with y: y = -√(25-x^2).
Step 2: Swap x and y: x = -√(25-y^2).
Step 3: Solve for y: Square both sides of the equation to remove the square root:
x^2 = (-√(25-y^2))^2
x^2 = 25 - y^2
Step 4: Rearrange the equation to isolate y^2:
y^2 = 25 - x^2
Step 5: Solve for y by taking the square root:
y = ±√(25 - x^2)
The inverse function of f(x) = -√(25-x^2) is given by: f^(-1)(x) = ±√(25 - x^2).