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f(x)=e^x+3+5, Need Domain(Which I couldn't find one) Where is the inverse?
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I did this problem, couldn't find a domain, but teacher states I should have one.
\[f(x)=e ^{(x+3)}+5\]Exponential functions have an unrestricted domain, so\[dom(f)=\mathbb{R}=(-\infty,\infty)\]
The function is one-to-one, so it has an inverse. To find the inverse, use the four step method. 1) Let f(x)=y, so\[f(x)=e ^{(x+3)}+5=y\]2) f^-1(x) is implied by\[x=e ^{(y+3)}+5\](here, we switch the x and y) 3) Solve for y\[x=e ^{y+3}+5\rightarrow x-5=e ^{y+3} \rightarrow \ln (x-5)=y+3\rightarrow \ln (x-5)-3=y\]4) Let y=f^-1(x), so \[\ln (x-5)-3=y=f ^{-1}(x)\]
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