If
and
are inverses of each other, then we have
![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)
Since
, we have
![f\left(f^{-1}(x)\right) = \dfrac1{f^{-1}(x)+2} = x](https://tex.z-dn.net/?f=f%5Cleft%28f%5E%7B-1%7D%28x%29%5Cright%29%20%3D%20%5Cdfrac1%7Bf%5E%7B-1%7D%28x%29%2B2%7D%20%3D%20x)
Solve for
:
![\dfrac1{f^{-1}(x)+2} = x \\\\ 1 = x\left(f^{-1}(x)+2\right) \\\\ 1 = x f^{-1}(x) + 2x \\\\ x f^{-1}(x) = 1 - 2x \\\\ f^{-1}(x) = \dfrac{1-2x}x \\\\ \boxed{f^{-1}(x) = \dfrac1x - 2}](https://tex.z-dn.net/?f=%5Cdfrac1%7Bf%5E%7B-1%7D%28x%29%2B2%7D%20%3D%20x%20%5C%5C%5C%5C%201%20%3D%20x%5Cleft%28f%5E%7B-1%7D%28x%29%2B2%5Cright%29%20%5C%5C%5C%5C%201%20%3D%20x%20f%5E%7B-1%7D%28x%29%20%2B%202x%20%5C%5C%5C%5C%20x%20f%5E%7B-1%7D%28x%29%20%3D%201%20-%202x%20%5C%5C%5C%5C%20f%5E%7B-1%7D%28x%29%20%3D%20%5Cdfrac%7B1-2x%7Dx%20%5C%5C%5C%5C%20%5Cboxed%7Bf%5E%7B-1%7D%28x%29%20%3D%20%5Cdfrac1x%20-%202%7D)
(provided that <em>x</em> ≠ 0 and <em>x</em> ≠ -2)
Answer:
a = p + 3
Step-by-step explanation:
a is whatever team scored 3 more runs than the Pirates, p is the Pirates.
We are given with the equation <span>f(x) = ax3 + bx2 + cx + d
Substituting, (3,11)
</span><span>11= 27a + 9b + 3c + d
</span><span>@(5, 9)
</span><span>9 = 125 a + 25 b + 5c + d
@</span><span>(4, 10)
</span><span>10 = 64 a + 16 b + 4c + d
@inflection point, second derivative is equal to zero
</span><span>f'(x) = 3ax2 + 2bx + c
</span>f''(x) = 6ax + 2b = 0
when x is 4, 24 a + 2b = 0 or 12a + b = 0.
There are 4 equations, 4 unknowns: answer is
<span>0.5 x^3 - 6x^2 + 22.5 - 24 = 0</span>
Answer:
8 pencils for every 4 students
Step-by-step explanation: