F (x ) = x^5 + 2x^3 + x − 1
a) What conditions must f (x ) satisfy in order to have an inverse function? Does is satisfy these conditions? why or why not?
b)Without finding an explicit function for for f^-1(x), find f^-1(3) and (f^-1)'(3)
a.
The conditions that f(x) must satisfy to have an inverse is:
1. f(x) must be a function
2. f(x) must be onto, i.e.
∀x ∈ dom f(x), f(x)=f(x1) <=> x=x1
condition 2 implies that f(x) must be strictly increasing or strictly decreasing.
This can be satisfied if f'(x)>0 ∀x∈dom f, or
f'(x)<0 ∀x∈dom f.
f(x) is a polynomial, therefore it is a function.
f'(x)=4x^4+6x^2+1 >0 ∀x∈R
therefore f(x) is strictly increasing, and has an inverse.
To find f-1(3), we need to solve for f(x)=3, which can be easily done by iteration or trial and error (in this case).
f-1'(x)=1/f'(x) if you recall that f(x) and f-1(x) are mirror images of each other about the line y=x, so the derivatives are related like tan(x) and cot(x).