1. ## inverse composite function

hi i need help on how to solve these questions

verify, algebraically, that $\displaystyle f(f{^-1}(x))$ = x

f(x) = $\displaystyle x^2$ -4

and

f(x) = $\displaystyle \frac{1}{x-2}$

2. Originally Posted by extraordinarymachine
hi i need help on how to solve these questions

verify, algebraically, that $\displaystyle f(f{^-1}(x))$ = x

f(x) = $\displaystyle x^2$ -4

and

f(x) = $\displaystyle \frac{1}{x-2}$
first you need to find the inverse of $\displaystyle x^2-4$

$\displaystyle y=x^2-4$

$\displaystyle x=y^2-4$

$\displaystyle x+4=y^2$

$\displaystyle \sqrt [2] {x+4}=\sqrt [2] {y^2}$

$\displaystyle \sqrt [2] {x}+2=y$

now we plug the inverse back into the original function

$\displaystyle (\sqrt [2] {x}+2)^2-4$

$\displaystyle x+4-4=x$

Can you follow the example to solve the second function

3. Originally Posted by extraordinarymachine
hi i need help on how to solve these questions

verify, algebraically, that $\displaystyle f(f{^-1}(x))$ = x

f(x) = $\displaystyle x^2$ -4
Unless you have some restriction on the possible values of x (on the domain) this function does not have an inverse!
RRH gets to $\displaystyle y^2= x+ 4$ and then says that $\displaystyle y= \sqrt{x+ 4}$ but there are two values of y whose square is x+ 4.

If the original function is "$\displaystyle f(x)= x^2- 4$ with domain $\displaystyle x\ge 0$", THEN $\displaystyle f^{-1}(x)= \sqrt{x+ 4}$ with domain $\displaystyle x\ge -2$.

If the original function is "$\displaystyle f(x)= x^2- 4$ with domain $\displaystyle x\le 0$, THEN $\displaystyle f^{-1}(x)= -\sqrt{x+ 4}$ with domain $\displaystyle x\ge -2$

RRH is also wrong when he writes $\displaystyle \sqrt{x+ 4}= \sqrt{x}+ 2$. For example, if x= 1, $\displaystyle \sqrt{1+ 4}= \sqrt{5}$ which is certainly not equal to $\displaystyle \sqrt{1}+ 2= 3$.

and

f(x) = $\displaystyle \frac{1}{x-2}$
Follow the pattern RRH used (which is correct). From $\displaystyle y= \frac{1}{x-2}$ (which has natural domain all real numbers except 2), the inverse function is given by $\displaystyle x= \frac{1}{y- 2}$, where I have swapped x and y. Now solve for y.

4. Originally Posted by HallsofIvy
Unless you have some restriction on the possible values of x (on the domain) this function does not have an inverse!
RRH gets to $\displaystyle y^2= x+ 4$ and then says that $\displaystyle y= \sqrt{x+ 4}$ but there are two values of y whose square is x+ 4.

If the original function is "$\displaystyle f(x)= x^2- 4$ with domain $\displaystyle x\ge 0$", THEN $\displaystyle f^{-1}(x)= \sqrt{x+ 4}$ with domain $\displaystyle x\ge -2$.

If the original function is "$\displaystyle f(x)= x^2- 4$ with domain $\displaystyle x\le 0$, THEN $\displaystyle f^{-1}(x)= -\sqrt{x+ 4}$ with domain $\displaystyle x\ge -2$

RRH is also wrong when he writes $\displaystyle \sqrt{x+ 4}= \sqrt{x}+ 2$. For example, if x= 1, $\displaystyle \sqrt{1+ 4}= \sqrt{5}$ which is certainly not equal to $\displaystyle \sqrt{1}+ 2= 3$.

Follow the pattern RRH used (which is correct). From $\displaystyle y= \frac{1}{x-2}$ (which has natural domain all real numbers except 2), the inverse function is given by $\displaystyle x= \frac{1}{y- 2}$, where I have swapped x and y. Now solve for y.
thank you for the correction I was thinking about that last night ... I have been sloppy with my math lately ... Again I apologize to the original poster to the original poster and to others in the MHF for my mistakes