Find a formula for the inverse of the function.

f(x)=1−(2/x^3) and f(x)=sqrt(x^2+7x)

∛(2/(1-x))

(-7±√(4x^2+49))/2
which branch depends on the domain of f(x)

thank you!

To find the inverse of a function, you need to follow these steps:

Step 1: Replace f(x) with y. This means rewriting the given function with y instead of f(x).

For the first function, f(x) = 1 - (2/x^3), we have:
y = 1 - (2/x^3).

For the second function, f(x) = sqrt(x^2 + 7x), we have:
y = sqrt(x^2 + 7x).

Step 2: Swap x and y variables. This means interchanging the x and y variables in the rewritten equations from Step 1.

For the first function, we have:
x = 1 - (2/y^3).

For the second function, we have:
x = sqrt(y^2 + 7y).

Step 3: Solve for y. Rearrange the equations obtained in Step 2 to solve for y, which will give you the inverse function.

For the first function, we need to solve for y:
x = 1 - (2/y^3).

Multiply both sides by y^3 to eliminate the fraction:
xy^3 = y^3 - 2.

Bring all terms to one side of the equation:
y^3 - xy^3 = -2.

Factor out y^3:
y^3(1 - x) = -2.

Divide both sides by (1 - x):
y^3 = -2/(1 - x).

Take the cube root of both sides to solve for y:
y = ( -2/(1 - x) )^(1/3).

This is the inverse function for the first given function.

For the second function, we need to solve for y:
x = sqrt(y^2 + 7y).

Square both sides of the equation to eliminate the square root:
x^2 = y^2 + 7y.

Rearrange the equation:
y^2 + 7y - x^2 = 0.

We have a quadratic equation. To solve for y, you can use the quadratic formula:
y = (-b ± sqrt(b^2 - 4ac)) / (2a).

In this case, a = 1, b = 7, and c = -x^2.

Substituting these values into the quadratic formula, we get:
y = (-7 ± sqrt(7^2 - 4(1)(-x^2))) / (2(1)).

Simplifying further, we have:
y = (-7 ± sqrt(49 + 4x^2)) / 2.

This is the inverse function for the second given function.