Note that f(x) as given is <em>not</em> invertible. By definition of inverse function,
![f\left(f^{-1}(x)\right) = x](https://tex.z-dn.net/?f=f%5Cleft%28f%5E%7B-1%7D%28x%29%5Cright%29%20%3D%20x)
![\implies f^{-1}(x)^3 - f^{-1}(x) + 2 = x](https://tex.z-dn.net/?f=%5Cimplies%20f%5E%7B-1%7D%28x%29%5E3%20-%20f%5E%7B-1%7D%28x%29%20%2B%202%20%3D%20x)
which is a cubic polynomial in
with three distinct roots, so we could have three possible inverses, each valid over a subset of the domain of f(x).
Choose one of these inverses by restricting the domain of f(x) accordingly. Since a polynomial is monotonic between its extrema, we can determine where f(x) has its critical/turning points, then split the real line at these points.
f'(x) = 3x² - 1 = 0 ⇒ x = ±1/√3
So, we have three subsets over which f(x) can be considered invertible.
• (-∞, -1/√3)
• (-1/√3, 1/√3)
• (1/√3, ∞)
By the inverse function theorem,
![\left(f^{-1}\right)'(b) = \dfrac1{f'(a)}](https://tex.z-dn.net/?f=%5Cleft%28f%5E%7B-1%7D%5Cright%29%27%28b%29%20%3D%20%5Cdfrac1%7Bf%27%28a%29%7D)
where f(a) = b.
Solve f(x) = 2 for x :
x³ - x + 2 = 2
x³ - x = 0
x (x² - 1) = 0
x (x - 1) (x + 1) = 0
x = 0 or x = 1 or x = -1
Then
can be one of
• 1/f'(-1) = 1/2, if we restrict to (-∞, -1/√3);
• 1/f'(0) = -1, if we restrict to (-1/√3, 1/√3); or
• 1/f'(1) = 1/2, if we restrict to (1/√3, ∞)
A, e, d are the answers :) hope I helped
The dividend on £1 is 15 p. Therefore, the dividend on £320.80 or 32080 pence (320.80 * 100) will be (32080*15)pence or 4812 pence, which is £48.12 (4812/100).
Step-by-step explanation:
Scores of Australia in all four matches starting from 1st 208, 192, 241, 203 respectively.
Similarly, scores of Pakistan in all four matches starting from 1st 102, 202, 360, 276 respectively.
The avg of Pakistan in one match = (102+202+360+276)/4 = 235
The avg of Australia in one match = (208+192+241+203)/4= 211
Since, the avg score of Pakistan is more than that of Australia, Pakistan performed better.
You have to add a 1 underneath it so it's reciprocal is 1