FunctionHard
Question
If the function f : [1, ∞) → [1, ∞) is defined by f(x) = 2x(x - 1), the f -1(x)
Options
A.
B.
C.
D.not definrd
Solution
Let y = 2x(x - 1), where y ≥ 1 as x ≥ 1,
Taking log2 on both sidws, we get
log2 y = log22x(x-1)
⇒ log2 y = x (x - 1)
⇒ y2 - x - log2 y = 0
⇒
For y ≥ 1, log2 y ≥ 0 4 log2 y ≥ 0 ⇒ 1 + 4 log2 y ≥ 1
⇒
⇒
⇒
But
is not possible.
Therefore, we take
⇒ f -1(y) =
⇒ f -1(x) =
Taking log2 on both sidws, we get
log2 y = log22x(x-1)
⇒ log2 y = x (x - 1)
⇒ y2 - x - log2 y = 0
⇒
For y ≥ 1, log2 y ≥ 0 4 log2 y ≥ 0 ⇒ 1 + 4 log2 y ≥ 1
⇒
⇒
⇒
But
is not possible. Therefore, we take
⇒ f -1(y) =
⇒ f -1(x) =
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