Question

Find the inverse funtion of p(t)=1+In(t)​

Answer 1

`\mathsf e^(t-1)`

Explanation:

Definition of an inverse function

`\textsf {A\:function\:g\:is\:the\:inverse\:of\:function\:f\:if\:for}\:y=f\left(x\right),\:\:x=g\left(y\right)\:`

In order to find the inverse of p(t) = 1 + ln(t)...

• Set y = 1 + ln(t)
• Replace t with y
  
  `\textsf t=1+\ln \left(y\right)`
• Solve for y
  
  `\textsf 1+\ln \left(y\right)=t`
• Subtract 1 from both sides
  
  `\textsf\ln \left(y\right)=t-1`
  `\textsf {$\ln \left(y\right)=t-1$}`
• Apply the log rule:
  `\mathsf{If}\:\log _a\left(b\right)=c\:\mathsf{then}\:b=a^c`
  
  `\ln \left(y\right)=t-1\quad`
  
  `\implies y=e^(t-1)`