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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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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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