Composite functions

**johett** · January 7th 2008, 07:48 AM

Let g(x)=x-3. Find a function f so that f(g(x))= $x^2$

Let f(x)= $x^2$ . Find a function g so that f(g(x))= $x^2+8x+16$

Thanks!

**topsquark** · January 7th 2008, 07:54 AM

Originally Posted by johett

Let g(x)=x-3. Find a function f so that f(g(x))= $x^2$

We know that
$f(x - 3) = x^2$

Assume we have a function f(x). Then f(x - 3) is a translation of this function 3 units to the right. And we know this function is $x^2$ . So if we translate the function f(x - 3) by 3 units to the left, we get f(x). So
$f(x) = (x + 3)^2$

-Dan

**topsquark** · January 7th 2008, 07:55 AM

Originally Posted by johett

Let f(x)= $x^2$ . Find a function g so that f(g(x))= $x^2+8x+16$

$f(g(x)) = x^2 + 8x + 16 = (x + 4)^2$

Since $f(x) = x^2$ then we know the argument of $f(g(x)) = (x + 4)^2$ must be $g(x) = x + 4$ .

-Dan

**Soroban** · January 7th 2008, 11:31 AM

Hello, johett!

Let $g(x)=x-3$
Find a function $f(x)$ so that: . $f(g(x)) \:=\: x^2$

We want: . $f(x-3) \:=\:x^2$

. . $f(x)$ must transform $x-3$ into $x^2.$

This can be done by: .adding 3 and squaring.

Therefore: . $f(x) \:=\:(x+3)^2$