If-f-x-2a-1-x-1-x-a-and-f-x-f-1-x-Find-a-2-3-

Question Number 208453 by hardmath last updated on 16/Jun/24

If f (x) = \frac{(2 a + 1) \cdot x + 1}{x - a} and f (x) = f^{- 1} (x)

Find : a^{2} + 3 = ?

Answered by efronzo1 last updated on 16/Jun/24

f^{- 1} (x) = \frac{ax + 1}{x - (2 a + 1)} = \frac{(2 a + 1) x + 1}{x - a}

(ax + 1) (x - a) = (x - (2 a + 1)) ((2 a + 1) x + 1)

{ax}^{2} - (a^{2} + 1) x - a = (2 a + 1) x^{2} + (1 - {(2 a + 1)}^{2}) x - (2 a + 1)

\Rightarrow a = 2 a + 1; a = - 1

\Rightarrow a^{2} + 3 = 4

Answered by MM42 last updated on 16/Jun/24

f (f (0)) = 0 \Rightarrow f (- \frac{1}{a}) = 0

\Rightarrow \frac{(2 a + 1) (- \frac{1}{a}) + 1}{- \frac{1}{a} - a} = 0

\Rightarrow \frac{2 a + 1}{a} = 1 \Rightarrow a = - 1 \Rightarrow a^{2} + 3 = 4 ✓

Commented by hardmath last updated on 16/Jun/24

thankyou dear professor