Composite functions

**koumori** · February 10th 2010, 12:24 PM

Am I working this problem properly?
If $f(x)=3x+1$ , and $g(x)=(x^2+5x)^{-\frac{1}{2}}$ , find $g(f(x))$ .

Working:
If $f(x)=3x+1$ , and $g(x)=(x^2+5x)^{-\frac{1}{2}}$ then, $g(f(x))$ = $((3x+1)^2+5(3x+1))^{-\frac{1}{2}}$ .

Am I correct, and is that the final answer?

**pickslides** · February 10th 2010, 12:27 PM

Originally Posted by koumori

Am I correct,

Yep

Originally Posted by koumori

and is that the final answer?

Expand inside the main bracket and group like terms to simplify

**koumori** · February 10th 2010, 03:10 PM

Originally Posted by pickslides

Yep

Expand inside the main bracket and group like terms to simplify

Ok...so the final answer looks like this:
$((9x^2+1)+(15x+5))^{-\frac{1}{2}}\rightarrow \frac{1}{\sqrt{(9x^2+1)+(15x+5)}}$ ?

**icemanfan** · February 10th 2010, 03:16 PM

Actually, what you should have found was that the expression simplifies to

$((9x^2 + 6x + 1) + (15x + 5))^{-\frac{1}{2}}$ ,

and that can be further simplified to

$(9x^2 + 21x + 6)^{-\frac{1}{2}}$ .

You can also take out a factor of 3:

$(3(3x^2 + 7x + 2))^{-\frac{1}{2}}$

**pickslides** · February 10th 2010, 05:49 PM

Originally Posted by koumori

Ok...so the final answer looks like this:
$((9x^2+1)+(15x+5))^{-\frac{1}{2}}\rightarrow \frac{1}{\sqrt{(9x^2+1)+(15x+5)}}$ ?

No, $(3x+1)^2 \neq 9x^2+1$

**koumori** · February 11th 2010, 11:00 AM

Originally Posted by icemanfan

Actually, what you should have found was that the expression simplifies to

$((9x^2 + 6x + 1) + (15x + 5))^{-\frac{1}{2}}$ ,

and that can be further simplified to

$(9x^2 + 21x + 6)^{-\frac{1}{2}}$ .

You can also take out a factor of 3:

$(3(3x^2 + 7x + 2))^{-\frac{1}{2}}$

Originally Posted by pickslides

No, $(3x+1)^2 \neq 9x^2+1$

I see! Because I should use F.O.I.L on $(3x+1)^2$ making it
$((9x^2 + 6x + 1) + (15x + 5))^{-\frac{1}{2}}$ . Then I have to do simple addition of like terms to net a result of $(9x^2 + 21x + 6)^{-\frac{1}{2}}$ . Then do as you said and factor everything by 3.
Finally, expand the exponent so $(9x^2 + 21x + 6)^{-\frac{1}{2}}$ becomes: $\frac{1}{\sqrt{(9x^2 + 21x + 6)}}$

Funny how I lost my head at this once the big words came out......

.

Thank you for your help guys!

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