Suppose h(x) = x 4 − x 7 . Evaluate (h −1 (2))4 − (h −1 (2))7 + 2

Question

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Answer 1

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