Calculus and Trigonometry

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fritzietheaTyro

Asked: April 15, 20212021-04-15T01:48:58+00:00 2021-04-15T01:48:58+00:00In: Calculus and Trigonometry, High School/Senior Secondary School

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acedstud · Accepted Answer · 2021-04-15T07:07:31+00:00

Best Answer

acedstud

To find the inverse of $f(x)=\ln(2x-3)+2$ , first, for simplicity sake, let’s substitute $y$ for $f(x)$ to get $y=\ln(2x-3)+2$ .

Next, subtract 2 from both sides to get $y-2=\ln(2x-3)$ .

Next, take the exponent of both sides to get $e^{y-2}=e^{\ln(2x-3)}=2x-3$ .

Next, add 3 to both sides to get $e^{y-2}+3=2x$ .

Next, divide through by 2 to get $\dfrac{1}{2}e^{y-2}+\dfrac{3}{2}=x\Rightarrow x=\dfrac{1}{2}e^{y-2}+\dfrac{3}{2}$ .

Finally, we will substitute $x$ for $y$ and $f^{-1}(x)$ for $x$ to get $f^{-1}(x)=\dfrac{1}{2}e^{x-2}+\dfrac{3}{2}$

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