Composite functions

• Jan 7th 2008, 08:48 AM
johett
Composite functions
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!
• Jan 7th 2008, 08:54 AM
topsquark
Quote:

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
• Jan 7th 2008, 08:55 AM
topsquark
Quote:

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
• Jan 7th 2008, 12:31 PM
Soroban
Hello, johett!

Quote:

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

Quote:

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

We want: . $f(g(x)) \:=\:(x+4)^2$

Therefore: . $g(x)\:=\:x+4$