InterviewSolution
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Let`f"":""NvecY`be a function defined as `f""(x)""=""4x""+""3`, where `Y""=""{y in N"":""y""=""4x""+""3`for some `x in N}`. Show that f is invertibleand its inverse is(1) `g(y)=(3y+4)/3`(2) `g(y)=4+(y+3)/4`(3) `g(y)=(y+3)/4`(4) `g(y)=(y-3)/4`A. `g(y)=(y+3)/(4)`B. `g(y)=(y-3)/(4)`C. `g(y)=(3y+4)/(3)`D. `g(y)=4+(y+3)/(4)` |
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Answer» Correct Answer - B For any `x, y in N` `f(x)=f(y) Rightarrow 4x+3=4y+3 Rightarrow x=y` `therefore` f is one-one Clearly, `Y=(y in N: y=4x+3"for some "x in N)="Range (f)"` `therefore` f:N-Y is onto. Thus, `f:N to Y` is a bijection and hence invertible. Let g be the inverse of f. Then. `fog (y)=y"for all "y inY` `Rightarrow f(g(y))=yRightarrow 4g(y)+3 =yRightarrow g(y)=(y-3)/(4)` |
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