What is the inverse of f(x)=x4+7 for x≥0 where function g is the inverse of function f?g(x)= 4√x+7, x≥−7g(x)= 4√x−7, x≥7g(x)=x√4−7, x≥0g(x)=x√4+7, x≥0
Accepted Solution
A:
[tex]\text{Consider the function, }\\
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f(x)=x^4+7, \ \ \text{ for }x\geq 0\\
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\text{let y=f(x). so in order to find the inverse first we interchange x and y.}\\
\text{so we have}\\
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x=y^4+7\\
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\text{and now we will solve for y again and that will give the inverse function.}\\
\text{so subtract 7 both sides, we get}[/tex][tex]x-7=y^4\\
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\text{now to get rid of exponent 4 from y, we take fourth root both sides.}\\
\text{so we get}\\
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\sqrt[4]{x-7}=y\\
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\text{hence the inverse function is }g(x)=\sqrt[4]{x-7}\\
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\text{we know that for a function its domain is the range of its inverse function}\\
\text{and the ragne of the function is domain of the inverse function.}\\
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\text{here observe that the range of f(x) is }f(x)\geq 7, \text{ domain of}[/tex][tex]\text{inerse function g(x) would be }x\geq 7\\
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\text{hence the inverse of the function is}\\
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g(x)=\sqrt[4]{x-7}, \ \ \ x\geq 7[/tex]