Suppose a function f has an inverse. If f(2)=6 and f(3)=7, find: f^-1 (f(3))

Mathematics

1 answer:

klemol [59]3 years ago

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Answer:

Step-by-step explanation:

The definition of an inverse function is ...

f^-1(f(x)) = x

Then ...

f^-1(f(3)) = 3

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Please someone help me..!!!!

Nikolay [14]

Answer: see proof below

Step-by-step explanation:

Use the Triple Angle Identity: sin 3A = 3sinA - 4sin³A

$\text{Given:}\quad \sin\bigg(\dfrac{\theta}{3}\bigg)=\dfrac{1}{2}\bigg(m+\dfrac{1}{m}\bigg)$

Proof LHS → RHS

LHS: sin Ф

Let A = Ф/3 sin 3A

Triple Angle Identity: 3sinA - 4sin³A

Substitute Ф/3 = A: 3sin(Ф/3) - 4sin³(Ф/3)

Substitute sin(Ф/3): $3\bigg(\dfrac{1}{2}\bigg(m+\dfrac{1}{m}\bigg)\bigg)-4\bigg(\dfrac{1}{2}\bigg(m+\dfrac{1}{m}\bigg)\bigg)^3$

$\text{Simplify:}\qquad \qquad \dfrac{3m}{2}+\dfrac{3}{2m}-4\bigg(\dfrac{1}{8}\bigg(m^3+3m+\dfrac{3}{m}+\dfrac{1}{m^3}\bigg)\bigg)\\\\.\qquad \qquad \qquad =\dfrac{3m}{2}+\dfrac{3}{2m}-\dfrac{m^3}{2}-\dfrac{3m}{2}-\dfrac{3}{2m}-\dfrac{1}{2m^3}\\\\.\qquad \qquad \qquad =-\dfrac{m^3}{2}-\dfrac{1}{2m^3}\\\\\text{Factor:}\qquad \qquad -\dfrac{1}{2}\bigg(m^3+\dfrac{1}{m^3}\bigg)$