Can someone explain how to do this please.

http://prntscr.com/bqt0kl

Question

Can someone explain how to do this please.

http://prntscr.com/bqt0kl

zepdrix · Answer

$\large\rm f=3x\qquad\qquad\qquad g=x+2$

For a, we'll take our g function and stick it inside of our f function.
What this means is, we're replacing the x in f, with the entire g function.
$\large\rm f\circ g=3g$
g is defined to be x+2, so we'll put that in place of our g,
with parenthesis around it, so the multiplication works out correctly,
$\large\rm f\circ g=3(x+2)$

zepdrix · Answer

Too confusing? What do you think? :)

mathguy5 · Answer

Ill try to review what you wrote down and tell you if I figured it out. Thank you :)

zepdrix · Answer

If you're able to understand this concept,
try applying the same idea to part b.

$\large\rm g=x+2$
In part b, we're sticking f into g.
$\large\rm g\circ f=f+2$

mathguy5 · Answer

@zepdrix Can you help me with this 1? http://prntscr.com/bqtn5g

zepdrix · Answer

$$\large\rm f=\color{orangered}{x}^2+1\qquad\qquad\qquad g=\sqrt{8-x}$$Ok this one is a little trickier, same idea though.
We'll take g and stuff it inside of our f function,$$\large\rm f\circ g=\color{orangered}{g}^2+1$$

zepdrix · Answer

And then we replace g with what it actually is ya?$$\large\rm f\circ g=(\sqrt{8-x})^2+1$$

mathguy5 · Answer

Yes I got -x+9 but my study guide says the answer is 9-x

zepdrix · Answer

Those are equivalent.
Maybe try to get a little stronger in arithmetic :)

$\large\rm 9-x\quad=9+-x$
You can rewrite subtraction as `addition of a negative`.
From there, we know that addition is `commutative`, we can add in any order.
So therefore, $\large\rm 9+-x=-x+9$

zepdrix · Answer

Part b is a bit more difficult.

When you have a composition of functions,
you have to respect the domain of the `inner function`
as well as the domain of the `resulting composite function`.

zepdrix · Answer

The resulting function is pretty straight forward,
-x+9 has no restrictions, all real numbers are ok.

But we need also look at the function which we "stuck inside",
and make sure it doesn't have any restrictions.

$\large\rm g=\sqrt{8-x}$
Do you understand how to determine the domain of this function g?

mathguy5 · Answer

Yes I do. Thank you for all the help :)

zepdrix · Answer

Ok good :)
So just keep in mind that your resulting composite function will still carry the restrictions from g(x).