Find the inverse of the function:
f(x)=-x²+16, x≥0
f(x)=5/x-3
x = -y^2 + 16
-y^2 = x-16
y^2 = -x + 16
y = sqrt (-x+16), x </= 16
do you mean
(5/x)-3
or
5/(x-3) ???
5/(x-3)
y = 5/(x-3)
x = 5/(y-3)
y x - 3 x = 5
y x = 3 x + 5
y = 3 + (5/x )
To find the inverse of a function, we need to swap the roles of x and y and solve for y.
For the first function, f(x) = -x² + 16, x ≥ 0:
1. Replace f(x) with y: y = -x² + 16.
2. Swap x and y: x = -y² + 16.
3. Solve for y:
Rearrange the equation:
y² = -x + 16.
Take the square root of both sides (taking the positive square root to match the initial condition x ≥ 0):
y = √(-x + 16).
So, the inverse of the function f(x) = -x² + 16, x ≥ 0, is given by f^(-1)(x) = √(-x + 16).
For the second function, f(x) = 5/x - 3:
1. Replace f(x) with y: y = 5/x - 3.
2. Swap x and y: x = 5/y - 3.
3. Solve for y:
Rearrange the equation:
x + 3 = 5/y.
Multiply both sides by y:
y(x + 3) = 5.
Expand:
xy + 3y = 5.
Solve for y:
3y = 5 - xy.
Divide both sides by 3:
y = (5 - xy)/3.
We can rearrange this equation to get y in terms of x, so:
y = (5/3) - (x/3).
Therefore, the inverse of the function f(x) = 5/x - 3 is given by f^(-1)(x) = (5/3) - (x/3).