Question

If f(x)=2x^3/5+7 ,find f^-1(x)?

Answer 1

`f^{-1}(x) = \root[3]{5/2(x-7)}`

Explanation:

Given a function `y=f(x)`, you can find its inverse `x = f^{-1}(y)` by solving the equation for `x`. By doing so, you change the roles of dependant and independant variables.

So, if we start from

`y = 2/5x^3+7`

We subtract `7` from both sides:

`y-7 = 2/5 x^3`

We multiply both sides by `5/2`:

`5/2(y-7)=x^3`

And finally extract the third root:

`\root[3]{5/2(y-7)} = x`

So, given `f(x)=2/5x^3+7`, we have `f^{-1}(x) = \root[3]{5/2(x-7)}`

As you can see here, the graphs of a function and its inverse are symmetrical with respect to `y=x`.