Suppose h(x) = x 4 − x 7 . Evaluate (h −1 (2))4 − (h −1 (2))7 + 2
Assumptions:
x 4 – x 7 actually means x^4 – x^7
h -1(2) actually means h^(-1)(2), i.e the value of the inverse function of h(x), at x = 2.
Inverse Functions
Let y = h(x) = some function of x.
If h() has an inverse, then x = h^(-1)(y)
So, h(x) = h(h^(-1)(y))
or, y = h(h^(-1)(y))
i.e. x = h(h^(-1)(x)) -------------------------------------- (1)
We have to evaluate E = (h^(-1)(2))^4 – (h^(-1)(2))^7 + 2
Let t = h^(-1)(2), then
E = t^4 – t^7 + 2, i.e.
E = h(t) + 2, or
E = h(h^(-1)(2)) + 2, now use (1)
E = 2 + 2
Answer: E = 4