Consider the function f- -9- - given by fx-5 x2-6 x-9 - Prove that f is invertible with f-1y-54-5 y-35 - -CBSE 2015-

We #160;have,fx=5x2+6x 9Let #160;y=5x2+6x 9=5x2+65x 95=5x2+2 #215;x #215;35+925 925 95=5x+352 925 95=5x+352 95 9=5x+352 545 #8658;y+545=5x+352 #8658 ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Special Integrals - 2

Consider the ...

Question

Consider the function f : R⁺ $\to [- 9, \infty]$ given by f(x) = 5x² + 6x $-$ 9. Prove that f is invertible with f^{$-$ 1}(y) = $\frac{\sqrt{54 + 5 y} - 3}{5}$ . [CBSE 2015]

Open in App

Solution

$We have, f (x) = 5 x^{2} + 6 x - 9 Let y = 5 x^{2} + 6 x - 9 = 5 (x^{2} + \frac{6}{5} x - \frac{9}{5}) = 5 (x^{2} + 2 \times x \times \frac{3}{5} + \frac{9}{25} - \frac{9}{25} - \frac{9}{5}) = 5 ({(x + \frac{3}{5})}^{2} - \frac{9}{25} - \frac{9}{5}) = 5 {(x + \frac{3}{5})}^{2} - \frac{9}{5} - 9 = 5 {(x + \frac{3}{5})}^{2} - \frac{54}{5} \Rightarrow y + \frac{54}{5} = 5 {(x + \frac{3}{5})}^{2} \Rightarrow \frac{5 y + 54}{25} = {(x + \frac{3}{5})}^{2} \Rightarrow \sqrt{\frac{5 y + 54}{25}} = x + \frac{3}{5} \Rightarrow x = \frac{\sqrt{5 y + 54}}{5} - \frac{3}{5} \Rightarrow x = \frac{\sqrt{5 y + 54} - 3}{5} Let g (y) = \frac{\sqrt{5 y + 54} - 3}{5} Now, f o g (y) = f (g (y)) = f (\frac{\sqrt{5 y + 54} - 3}{5}) = 5 {(\frac{\sqrt{5 y + 54} - 3}{5})}^{2} + 6 (\frac{\sqrt{5 y + 54} - 3}{5}) - 9 = 5 (\frac{5 y + 54 + 9 - 6 \sqrt{5 y + 54}}{25}) + (\frac{6 \sqrt{5 y + 54} - 18}{5}) - 9 = \frac{5 y + 63 - 6 \sqrt{5 y + 54}}{5} + \frac{6 \sqrt{5 y + 54} - 18}{5} - 9 = \frac{5 y + 63 - 18 - 45}{5} = y = I_{Y}, Identity function Also, g o f (x) = g (f (x)) = g (5 x^{2} + 6 x - 9) = \frac{\sqrt{5 (5 x^{2} + 6 x - 9) + 54} - 3}{5} = \frac{\sqrt{25 x^{2} + 30 x - 45 + 54} - 3}{5} = \frac{\sqrt{25 x^{2} + 30 x + 9} - 3}{5} = \frac{\sqrt{{(5 x + 3)}^{2}} - 3}{5} = \frac{5 x + 3 - 3}{5} = x = I_{X}, Identity function So, f is invertible . Also, f^{- 1} (y) = g (y) = \frac{\sqrt{5 y + 54} - 3}{5}$

Suggest Corrections

Similar questions

Q. Consider

f : R^{+} \to [- 9, \infty]

given by

f (x) = 5 x^{2} + 6 x - 9

. Prove that

f

is invertible with

f^{'} (y) = (\frac{\sqrt{54 + 5 y} - 3}{5})

Consider f: R₊ → [−5, ∞) given by f(x) = 9x² + 6x − 5. Show that f is invertible with.

Q. Consider

f : R^{+} \in [- 5, \infty)

given by

f (x) = 9 x^{2} + 6 x - 5

. Show that

f

is invertible with

f^{- 1} (y) = (\frac{(\sqrt{y + 6}) - 1}{3})

Consider f: R₊→ [4, ∞) given by f(x) = x² + 4. Show that f is invertible with the inverse f⁻¹ of given f by, where R₊ is the set of all non-negative real numbers.

Consider f: R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.

Join BYJU'S Learning Program

Consider the function f : R+ →-9,∞ given by f(x) = 5x2 + 6x - 9. Prove that f is invertible with f-1(y) = 54+5y-35. [CBSE 2015]

Consider the function f : R⁺ $\to [- 9, \infty]$ given by f(x) = 5x² + 6x $-$ 9. Prove that f is invertible with f^{$-$ 1}(y) = $\frac{\sqrt{54 + 5 y} - 3}{5}$ . [CBSE 2015]