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Write explicit functions of `y` defined by the following equations and also find the domains of definitions of the given implicit functions: `x+|y|=2y` (b) `e^y-e^(-y)=2x` `10^x+10^y=10` (d) `x^2-sin^(-1)y=pi/2`

Text Solution

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(i) `x+|y|=2y`
If `y ge 0, ` we have ` x+y=2yimpliesx`
`implies y=x,x ge 0`
If `y lt 0, ` we have ` x-y=2yimplies y=(x)/(3)`
`implies y=(x)/(3), x lt 0`
Hence, `y={((x)/(3)", "x lt 0),(x", "x ge 0):}`
(ii) `e^(y) -e^(-y)=2x`
`implies e^(2y)-2xe^(y)-1=0 " " `(Multiplying by `e^(y)`)
`implies e^(y)=(2x+-sqrt(4x^(2)+4))/(2)=x+-sqrt(x^(2)+1)`
`implies e^(y)=x+sqrt(x^(2)+1)" (as "sqrt(x^(2)+1) gtx, " then " x-sqrt(x^(2)+1) lt 0, " which is not possible)" `
`implies y=log_(e)(x+sqrt(x^(2)+1))`
(iii) `10^(x)+10^(y)=10`
`implies 10^(y)=10-10^(x)`
`implies y=log_(10)(10-10^(x))`
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