How do you show that $f(x)=9-x^2$ and $g(x)=sqrt(9-x)$ are inverse functions algebraically and graphically?

Question

6235 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-01-31T17:54:30

See proof below

We calculate the composition of the functions

$f(x)=9-x^2$

$g(x)=sqrt(9-x)$

$f(g(x))=f(sqrt(9-x))=9-(sqrt(9-x))^2$

$=9-(9-x)=x$

$g(f(x))=g(9-x^2)=sqrt(9-(9-x^2))$

$sqrtx^2=x$

Therefore, $f(x)$ and $g(x)$ are inverses

$f(x)=g^-1(x)$

and $g(x)=f^-1(x)$

Graphically, $f(x)$ and $g(x)$ are symmetrics wrt $y=x$

graph{(y-9+x^2)(y-sqrt(9-x))(y-x)=0 [-1.78, 43.84, -9.37, 13.44]}

How do you show that f(x)=9-x^2f(x)=9-x^2 and g(x)=sqrt(9-x)g(x)=sqrt(9-x) are inverse functions algebraically and graphically?