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Sagot :
[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
[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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