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How do you find the inverse of $y = \sqrt{x - 4}$ $y =sqrt(x-4)$ ?

1 Answer

Dec 9, 2015

$x = y^{2} + 4$ $x=y^2+4$
or if you wish to keep $y$ $y$ as the dependent variable:
$y = x^{2} + 4$ $y=x^2+4$

Explanation:

Method 1:
$y = \sqrt{x - 4}$ $y=sqrt(x-4)$

$\Rightarrow y^{2} = x - 4$ $rArr y^2 = x-4$

$\Rightarrow y^{2} + 4 = x XXx or XXX x = y^{2} + 4$ $rArr y^2+4 = x color(white)("XXx")orcolor(white)("XXX")x=y^2+4$

Method 2: (perhaps more complex but more technically accurate)
Let
$XXX f (x) = y$ $color(white)("XXX")f(x)=y$
and
$XXX ¯ ¯¯¯¯¯ ¯ f (x)$ $color(white)("XXX")barf (x)$ be the inverse of $f (x)$ $f(x)$

By definition of inverse:
$XXX f (¯ ¯¯¯¯¯ ¯ f (x)) = x$ $color(white)("XXX")f(barf(x)) = x$

But since
$XXX f (x) = \sqrt{x - 4}$ $color(white)("XXX")f(x) = sqrt(x-4)$
$\Rightarrow$ $rArr$
$XXX f (¯ ¯¯¯¯¯ ¯ f (x)) = \sqrt{¯ ¯¯¯¯¯ ¯ f (x) - 4}$ $color(white)("XXX")f(barf(x))= sqrt(barf(x)-4)$

So
$XXX \sqrt{¯ ¯¯¯¯¯ ¯ f (x) - 4} = x$ $color(white)("XXX")sqrt(barf(x)-4) = x$

$XXX ¯ ¯¯¯¯¯ ¯ f (x) - 4 = x^{2}$ $color(white)("XXX")barf(x)-4 = x^2$

$XXX ¯ ¯¯¯¯¯ ¯ f (x) = x^{2} + 4$ $color(white)("XXX")barf(x)=x^2+4$

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