How do you find f^{-1}(c) if f(x)= 5x + 8 x^{11}; c = -13?

2 Answers
Nov 11, 2016

f(x)=8x^11+5x

Let
n = f^(-1)(-13).
f(n)=-13

We have to check where f(x) is equal to -13.
-13=8x^11+5x

After graphing, I found that f(x) is equal to -13 at x=-1.

Nov 11, 2016

f^(-1)(-13)=-1

Explanation:

In general the given function is in the form y=f(x) and its inverse is f^(-1)(y)=x, provided that f(x) is always crescent or decrescent for each value in the domain. This is just the case, being f'(x)=5+11x^(10) always positive for any x in R.

checked this, the problem of finding f^(-1)(-13)=x can be seen in an equivalent way as the one of finding the value of x whose image is -13.

To find it, it is enough to solve the equation 8x^(11)+5x-13=0.
This has one root in x=-1 so that at this point in the domain corresponds the value y=-13 in the image. Being f(-1)=13 and verified its monotonicity, we can concude that f^(-1)(-13)=-1