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If [tex]${data-answer}amp;f(x)=\frac{19}{x^{2}} \\[/tex] then the inverse function exists [tex]${data-answer}amp;f^{-1}(x)=\sqrt{\frac{19}{x}}[/tex].
What is the meaning of inverse function?
An inverse function in mathematics exists function which "reverses" the another function.
Let f(x) = y, then the inverse function, [tex]$x=f^{-1}(y)$[/tex]
[tex]${data-answer}amp;f(x)=\frac{19}{x^{2}} \\[/tex]
[tex]${data-answer}amp;y=\frac{19}{x^{2}} \\[/tex]
[tex]${data-answer}amp;x^{2}=\frac{19}{y} \\[/tex]
simplifying the equation, we get
[tex]${data-answer}amp;x=\sqrt{\frac{19}{y}} \\[/tex]
[tex]${data-answer}amp;x^{-1}=f^{-1}(y)=\sqrt{\frac{19}{y}} \\[/tex]
[tex]${data-answer}amp;f^{-1}(y)=\sqrt{\frac{19}{y}},[/tex] then [tex]${data-answer}amp;f^{-1}(x)=\sqrt{\frac{19}{x}}[/tex].
If [tex]${data-answer}amp;f(x)=\frac{19}{x^{2}} \\[/tex] then the inverse function exists [tex]${data-answer}amp;f^{-1}(x)=\sqrt{\frac{19}{x}}[/tex].
To learn more about inverse function refer to:
https://brainly.com/question/11735394
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