Question

. Find the inverse of each function.

Answer 1

Answer:

a. $f^{-1}(x)=\dfrac{x+3}{2}$

b. $g^{-1}(x) =e^{\frac{x}{2}}+1$

c. $h^{-1}(x)=\dfrac{e^{y}}{e^{y}-1}$

d. $k^{-1}(x)=-\dfrac{2log(5-y)}{log(3)}$

Step-by-step explanation:

Here is the procediment for each case.

a. $f(x)=2x-3$ as we know $f(x)=y$ then $y=2x-3$ finding the expresion of x, we find the inverse of the function, then:

$y=2x-3\\y+3=2x\\\\\dfrac{y+3}{2}=x$

Then:

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

b. $g(x)=2log(x-1)$ for this we have to remember that the inverse function of the log is the exp, then:

$g(x) = 2 log(x-1)\\y=2log(x-1)\\\\\\\dfrac{y}{2}=log(x-1)\\\\e^{\frac{y}{2}}=e^{log(x-1)}\\\\e^{\frac{y}{2}}=x-1\\e^{\frac{y}{2}}+1=x$

Then the inverse function is:

$g^{-1}(x) =e^{\frac{x}{2}}+1$

c. In this case we have to also remember the relation between the e and the ln, then:

$h(x) = ln(x)-ln(x-1)$  with the properties of the ln we have:

$h(x) = ln(x)-ln(x-1)=ln\left(\dfrac{x}{x-1}\right)$ now finding the inverse function we have:

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

then:

$h^{-1}(x)=\dfrac{e^{y}}{e^{y}-1}$

d. in the last one we have: $k(x) =5-3^{-x/2}$ then:

$k(x) =5-3^{-x/2}\\y =5-3^{-x/2}\\y-5=-3^{-x/2}\\5-y=3^{-x/2}\\log(5-y)=log(3^{-x/2})\\log(5-y)=-\frac{x}{2}log(3)\\-2log(5-y)=xlog(3)\\-\dfrac{2log(5-y)}{log(3)}=x$

then:

$k^{-1}(x)=-\dfrac{2log(5-y)}{log(3)}$