# Composite functions

• Jan 7th 2008, 07:48 AM
johett
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!
• Jan 7th 2008, 07:54 AM
topsquark
Quote:

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
• Jan 7th 2008, 07:55 AM
topsquark
Quote:

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
• Jan 7th 2008, 11:31 AM
Soroban
Hello, johett!

Quote:

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\$

Quote:

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\$