Question

How do you determine if the function is a one-to-one function and find the formula of the inverse given `f(x) = 5x^3 - 7`?

Answer 1

See below

Explanation:

Let `f:RR->RR, f(x)=5x^3-7` be a function from `RR` to `RR`.

We want to prove that `f` is injective over `RR` and then find its inverse. But we will do it the other way around; ie., we will find the inverse of `f` and this will be sufficient to prove that `f` is injective.

`f(x)=5x^3-7`

so `f(x)+7=5x^3`

so `(f(x)+7)/5=x^3`

so `((f(x)+7)/5)^(1/3)=x`

so `f^-1(x)=((x+7)/5)^(1/3)`

Now we check that `f(f^-1(x))=x` and `f^-1(f(x))=x`

`f(f^-1(x))=5(((x+7)/5)^(1/3))^3-7=cancel5 *(x+cancel7)/cancel5-cancel7=x`

`f^-1(f(x))=((5x^3-7+7)/5)^(1/3)=((5x^3)/5)^(1/3)=x`

So `f^-1` is the inverse of `f`. Now, since `f` has an inverse, it must be bijective, and so it must be injective.