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Consider `f: R->R`given by `f(x) = 4x + 3`. Show that f is invertible. Find the inverse of f. |
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Answer» In `f: R to R, f(x) = 4x + 3` Let `x, y in R and f(x) = f(y)` `rArr 4x + 3 = 4y + 3` `rArr 4x = 4y rArr x =y ` ` therefore f` is one-one. Again, let `f(x) = y ` where `y in R` `rArr " " 4x + 3 = y rArr 4x = y - 3 ` `rArr " "x = ( y -3)/( 4) ` Now for each ` y in R, x = (y - 3)/(4) in R ` in such that `f(x) = f((y -3)/(4))= 4((y-3)/( 4)) + 3 = y` `therefore f ` is onto. Therefore, `f` is one-one onto function ` rArr f` is invertible. `therefore f^(-1) : R to R ` is defined as `f^(-1)(y) = (y-3)/(4)`. |
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