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Let `f: N -> R`be a function defined as `f(x)=4x^2+12 x+15.`Show that `f: N -> S,`where `S`is the range of `f`is invertible. Also find the inverse of `f` |
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Answer» a function is invertible only one-one and onto `x_1,x_2 in R` `f(x_1)=f(x_2)` `x_1=x_2` `4x_1^2+12x_1+15=4x_2^2+12x_2+15` `4(x_1^2-x_2^2)+12(x_1-x_2)=0` `(x_1-x_2)(x_1+x_2)+3(x_1-x_2)=0` `(x_1-x_2)[x_1+x_2+x_3]=0` This is one one function function has range=co-domain f(x) is one one and onto function `f(x)=4x^2+12x+15` `f(x)=(2x)^2*(2x)(3)+3^2+6` `y=f(x)=(2x+3)^2+6` `y=(2x+3)^2+6` `(y-6)=(2x+3)^2` `(2x+3)=sqrt(y-6` `x=(sqrt(y-6)-3)/2` `f^(-1)(Y)=(sqrt(y-6)-3)/2`. |
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