1. ## Composite functions

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

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

Thanks!

2. Originally Posted by johett
Let g(x)=x-3. Find a function f so that f(g(x))=$\displaystyle x^2$
We know that
$\displaystyle f(x - 3) = x^2$

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 $\displaystyle x^2$. So if we translate the function f(x - 3) by 3 units to the left, we get f(x). So
$\displaystyle f(x) = (x + 3)^2$

-Dan

3. Originally Posted by johett
Let f(x)=$\displaystyle x^2$. Find a function g so that f(g(x))=$\displaystyle x^2+8x+16$
$\displaystyle f(g(x)) = x^2 + 8x + 16 = (x + 4)^2$

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

-Dan

4. Hello, johett!

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

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

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

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

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

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

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

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