Respuesta :
Given the function:
[tex]f(x)=2^{(x+1)}-3[/tex]You can find its inverse as follows:
1. Rewrite the function using:
[tex]y=f(x)[/tex]Then:
[tex]y=2^{(x+1)}-3[/tex]2. Solve for "x":
- Add 3 to both sides of the equation:
[tex]y+3=2^{(x+1)}-3+3[/tex][tex]y+3=2^{(x+1)}[/tex]- Apply logarithm to both sides:
[tex]log(y+3)=log(2)^{(x+1)}[/tex]- Apply the Power Property for Logarithms:
[tex]log(m)^n=nlog(m)[/tex]Then:
[tex]log(y+3)=(x+1)log(2)^[/tex]- Divide both sides of the equation by:
[tex]log(2)[/tex]You get:
[tex]\frac{log(y+3)}{log(2)}=\frac{x+1}{log(2)}[/tex][tex]\frac{log(y+3)}{log(2)}=x+1[/tex]- Subtract 1 from both sides:
[tex]\frac{log(y+3)}{log(2)}-1=x+1-1[/tex][tex]\frac{log(y+3)}{log(2)}-1=x[/tex]3. Swap variables:
[tex]\frac{log(y+3)}{log(2)}-1=x+1-1[/tex][tex]y=\frac{log(x+3)}{log(2)}-1[/tex]4. Rewrite it in this form:
[tex]f^{-1}(x)=\frac{log(x+3)}{log(2)}-1[/tex]Hence, the answer is:
[tex]f^{-1}(x)=\frac{log(x+3)}{log(2)}-1[/tex]