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\( F:[0, \infty) \rightarrow[9, \infty) \) defined as \( F(x)=4 x^{2}+4 x \) find \( F^{-1}(x) \) if exist. |
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Answer» Let F(x) = 4x2 + 4x = y ⇒ x2 + x = y/4 ⇒ (x + 1/2)2 - 1/4 = y/4 ⇒ (x + 1/2)2 = y/4 + 1/4 = \(\frac{y+1}4\) ⇒ x + 1/2 = \(\sqrt{\frac{y+1}4}\) (\(\because\) Rang of F is [9, \(\infty\)] which is positive) ⇒ x = \(\sqrt{\frac{y+1}4}-\frac12\) ⇒ F-1(y) = \(\frac{\sqrt{y+1}-1}2\) (\(\because\) F(x) = y ⇒ x = F-1(y)) \(\therefore\) F-1 = \(\frac{\sqrt{x+1}-1}2\) |
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