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Assume that $$f$$ is one $$-$$ to $$-$$ one function
(a) If $$f(6)=17$$ what is $$f^{-1}(17)$$
(b) If $$f^{-1}(3)=2$$ what is $$f(2)$$

(a) $$f(6)=17 \rightarrow f^{-1}(17)=? ?$$

$$f^{-1}(17)=6$$

(b) $$f^{-1}(3)=2 \rightarrow f(2)=? ?$$

$$f(2)=3$$

Find a formula for the inverse of the function

$$f(x)=\frac{4 x-1}{2 x+3}$$

(1) $$y=\frac{4 x-1}{2 x+3}$$

(2) $$2 y x+3 y=4 x-1$$

$$2 y x-4 x=-3 y-1$$

$$x(2 y-4)=-(3 y+1) \quad \div 2 y-4$$

$$x=\frac{-(3 y+1)}{2 y-4}$$

(3) $$y=\frac{-(3 x+1)}{2 x-4}$$

(4) $$f^{-1}(x)=\frac{-(3 x+1)}{2 x-4}$$

Let $$f(x)=\ln \left(e^{x}-1\right)$$
(a) Find the domain of $$f$$
(b) Find $$F^{-1}$$ and its domain

(a) $$f(x)=\ln \left(e^{x}-1\right)$$

$$e^{x}-1>0$$

$$e^{x}>1$$

$$\ln e^{x}>\ln (1)$$

$$x>0$$

Domain $$(0,+\infty)$$

(b) $$F^{-1}=? ?$$

$$f(x)=\ln \left(e^{x}-1\right)$$

$$y=\ln \left(e^{x}-1\right)$$

$$e^{y}=e^{\ln \left(e^{x}-1\right)}$$

$$e^{y}=e^{x}-1$$

$$e^{x}=e^{y}+1$$

$$\ln e^{x}=\ln \left(e^{y}+1\right)$$

$$x=\ln \left(e^{y}+1\right)$$

$$y=\ln \left(e^{x}+1\right)$$

$$F^{-1}(x)=\ln \left(e^{x}+1\right)$$

$$e^{x}+1>0$$

$$e^x>-1$$

$$\ln e^{x} > \ln-1$$

$$x>\ln (-1)$$

Drmain $$(-\infty,+\infty)$$

Let $$f(x)=\frac{x}{1+2 x}$$
(a) Find the domain of $$f$$
(b) Find $$F^{-1}(x)$$
(c) Find the range of $$F$$

(a) $$f(x)=\frac{x}{1+2 x}$$

$$1+2 x=0 \quad 2 x=-1 \quad x=-\frac{1}{2}$$

Domain $$=R /\left\{-\frac{1}{2}\right\}$$

(b) $$F^{-1}(x) ?$$

$$y=\frac{x}{1+2 x}$$

$$y+2 y x=x$$

$$2 y x-x=-y$$

$$x(2 y-1)=-y$$

$$x=\frac{-y}{2 y-1}$$

$$y=\frac{-x}{2 x-1}$$

$$F^{-1}(x)=\frac{-x}{2 x-1}$$

$$D F^{-1}=R /\left\{\frac{1}{2}\right\}$$

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