Question

What is the inverse of f(x)=(x−5)2 for x≥5 where function g is the inverse of function f? g(x)=x−5−−−−√, x≥5g(x)=x√+5, x≥0g(x)=x+5−−−−√, x≥−5g(x)=x√−5, x≥0

Answer 1

To solve for the inverse of the function

`f(x)=(x-5)^2\text{ for x }\ge\text{ 5}`

`\mathrm{A\: function\: g\: is\: the\: inverse\: of\: function\: f\: if\: for}\: y=f\mleft(x\mright),\: \: x=g\mleft(y\mright)`
`\begin{gathered} y=\mleft(x-5\mright)^2 \\ \text{replace x with y} \\ x=(y-5)^2 \\ solve\text{ for y} \end{gathered}`
`\begin{gathered} x=(y-5)^2 \\ \text{square root both side} \\ \sqrt[]{x}=\sqrt[]{(y-5)^2} \\ √(x)=y-5 \\ y=\pm\sqrt[]{x}+5 \end{gathered}`

The function g(x) = y, Hence

`g(x)=\sqrt[]{x}+5,\text{ }for\text{ x}\ge0`

Therefore the correct answer is Option B