y=(x^1/2 - 3 )^2 + (x^1/2 - 3) +5

[math1325]

write y=(x^1/2 - 3 )^2 + (x^1/2 - 3) +5 as the composition of two function. Hint: let one function be a pattern that repeats.

This is what I have so far...
f(x)= 1x^2 + 1x +5x
= x^2+x+5x
= x^2+6x
g(x)= (x^1/2 - 3)

y=f(g(x))
= x^2+6x
= (x^1/2 - 3)^2 + 6(x^1/2 - 3)
= (x^1/2 - 3)^2 + 6x^1/2 - 18
= (x^1/2 - 3) (x^1/2 - 3)
= (x-3x^1/2 - 3x^1/2 + 9) + 6x^1/2 - 18
= x - 6x^1/2 + 9 + 6x^1/2 - 18
= x-9

Please help...thank you.

 

[tkhunny]

y = x^2+6x

Where did the "+5" go? Anyway, you tried to write f(g(x)) and managed only f(x).

= (x^1/2 - 3)^2 + 6x^1/2 - 18
= (x^1/2 - 3) (x^1/2 - 3)

Again, just a piece of it. Where did the rest go?

= (x-3x^1/2 - 3x^1/2 + 9) + 6x^1/2 - 18

There it is? Too bad the +5 didn't come back just as magically.

Try it again and be MUCH more careful. Don't try to do three things at once.

 

[soroban]

Hello, math1325!

\(\displaystyle \text{Write }y\;=\;(x^{\frac{1}{2}}\,-\,3)^2\,+\,(x^{\frac{1}{2}}\,-\,3)\,+\,5\text{ as the composition of two functions.}\)

\(\displaystyle \text{Look at what we have: }\,y\:=\:u^2\,+\,u\,-\,\,5\text{ where }\,u\:=\:x^{\frac{1}{2}}\,-\,3\)

\(\displaystyle \text{This should suggest: }\,f(x)\:=\:x^2\,+\,x\,-\,5\.\text{ and }\,g(x)\:=\:x^{1/2}\,-\,3\)

\(\displaystyle \text{And we have: }\,y\;=\;f(g(x))\)

 

[math1325]

This is what I was able to solve so far.....but I think I still need help


f(x)= x^2 + x + 5
than I replaced x with g(x) which is x^1/2 -3
= (x^1/2 -3)^2 +(x^1/2 -3) +5
= [x - 3x^1/2 - 3x^1/2 +9] + (x^1/2 -3) +5
= x - 5x^1/2 +11

for g[f(x)]
=x^1/2 -3
=(x^2 + x +5) ^1/2 -3
= don't have any idea how to go about solving any further

 

[skeeter]

This is what I was able to solve so far.....but I think I still need help


f(x)= x^2 + x + 5
than I replaced x with g(x) which is x^1/2 -3
= (x^1/2 -3)^2 +(x^1/2 -3) +5

you are "done" with the step above ... the question (as you stated it originally) just asked you to express the original function as the composition of two functions. you don't have to do anything else.

= [x - 3x^1/2 - 3x^1/2 +9] + (x^1/2 -3) +5
= x - 5x^1/2 +11

for g[f(x)]
=x^1/2 -3
=(x^2 + x +5) ^1/2 -3
= don't have any idea how to go about solving any further