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[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]

  • Option A is correct

[tex]\textsf{ \underline{\underline{Steps to solve the problem} }:}[/tex]

Find inverse of given function :

[tex]\qquad❖ \: \sf \:y = \sqrt{x} + 7[/tex]

[tex]\qquad❖ \: \sf \: \sqrt{x} = y - 7[/tex]

[tex]\qquad❖ \: \sf \:x = (y - 7) {}^{2} [/tex]

Next, replace x with f-¹(x) and y with x ~

[tex]\qquad❖ \: \sf \:f {}^{ - 1} (x) = (x - 7) {}^{2} [/tex]

we got our inverse function.

Condition : x should be greater or equal to 7

because we will get same value of y for different x if we also include values less than 7.

[tex] \qquad \large \sf {Conclusion} : [/tex]

  • Correct option is A
Kailes

[tex] \huge\mathbb{ \underline{SOLUTION :}}[/tex]

Given:

[tex]\longrightarrow\bold{f(x)= \sqrt{7}}[/tex]

To solve for the inverse of a function we begin by re-writing the function as an equation in terms of y.

[tex]\bold{Becomes,}[/tex]

Next step we switch sides for x and y variables and then solve for the y variable as shown below,

[tex]\longrightarrow\sf{y= \sqrt{x}+7}[/tex]

[tex]\bold{Then,}[/tex]

[tex]\longrightarrow\sf{x= \sqrt{y}+7}[/tex]

[tex]\small\bold{Solve \: for \: y \: and \: subtract \: 7 \: from \: the \: both \: }[/tex] [tex]\bold{sides,}[/tex]

[tex]\longrightarrow\sf{x-7= \sqrt{y}}[/tex]

[tex]\small\bold{Square \: both \: sides }[/tex]

[tex]\sf{(x-7)^2=(\sqrt{y})^2}[/tex]

[tex]\sf{(x-7)^2=y}[/tex]

We now re-write in function notation. Take note however that this is the inverse:

[tex]\bold{Where \: y}[/tex] [tex]\sf{=(x-7)^2 }[/tex]

[tex]\longrightarrow\sf{y= (x-7)^2 }[/tex]

[tex]\huge\mathbb{ \underline{ANSWER:}}[/tex]

[tex]\large\boxed{\sf A. \: \: f^{-1}(x)= (x − 7)^2 , \: for \: \underline > 7 }[/tex]

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