Find a formula for the inverse of the function
f(x)=9−4/x2, x>0.
original:
y = 9 - 4/x^2
inverse:
x = 9 - 4/y^2
solving that for y:
xy^2 = 9y^2 - 4
xy^2 - 9y^2 = -4
y^2(x -9) = -4
y^2 = -4/(x-9)
y^2 = 4/(9-x)
y = ± √(9-x)
the new equation is NOT a function
To find the formula for the inverse of a function, we need to swap the roles of "x" and "y" and solve for "y". Let's do that for the given function:
1. Start with the original function:
f(x) = 9 - 4/x^2, x > 0
2. Swap "x" and "y" in the equation:
x = 9 - 4/y^2
3. Solve the equation for "y":
x - 9 = -4/y^2
Multiplying both sides by -1:
9 - x = 4/y^2
Taking the reciprocal of both sides:
1/(9 - x) = y^2/4
Taking the square root of both sides:
sqrt(1/(9 - x)) = y/2
4. Simplify the equation:
y = ± 2 * sqrt(1/(9 - x))
So, the formula for the inverse of the given function f(x) = 9 - 4/x^2, x > 0 is:
f^(-1)(x) = ± 2 * sqrt(1/(9 - x))