Science:Math Exam Resources/Courses/MATH104/December 2016/Question 05 (b)

MATH104 December 2016

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Question 05 (b)
Let $f(x)=x+e^{x}$ . Find the equation of the tangent line of $y=f^{-1}(x)$ at $x=1$ . Recall that $f^{-1}(x)$ is the inverse function of $f(x)$ and its derivative can be calculated using formula ${\frac {df^{-1}}{dx}}={\frac {1}{f'(f^{-1}(x))}}$ .

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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
In order to get the tangent line, we need the point $(1,f^{-1}(1))$ , and the derivative of $f^{-1}(x)$ at -1. Recall the relationship between $f^{-1}(1)$ and $f(x)=1$ .

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Solution
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Notice that $f(0)=1$ , so $f^{-1}(1)=0$ . It remains to find the slope of the tangent line. $y'={\frac {1}{f'(f^{-1}(x))}}={\frac {1}{f'(0)}}$ . So, we need $f'(x)=1+e^{x}\rightarrow f'(0)=2\rightarrow {\text{slope}}={\frac {1}{2}}$ . Therefore, answer: $\color {blue}y={\frac {1}{2}}(x-1)$ .