I have an exam tomorrow and I'm struggling with composite functions, particularly finding 2 to 3 simpler ones to describe one and ones with fractions.

Question

lifeisadangerousgame · Answer

$$f(x) = \frac{ x + 2 }{ x - 3}; g(x) = \frac{ 1 }{ x + 1 }$$ Find formulas for f(g(x)) and g(f(x))That's one, if you wouldn't mind showing me how to do it step by step I appreciate it.

dumbcow · Answer

its all about replacing the variable "x" with whatever the other function is
---> f(g) means that the variable is g(x) not "x"
$$f(g(x)) = \frac{g(x) + 2}{g(x) - 3} = \frac{\frac{1}{x+1} +2}{\frac{1}{x+1} -3}$$

dumbcow · Answer

$$g(f(x)) = \frac{1}{f(x) +1} = \frac{1}{\frac{x+2}{x-3}+1}$$

lifeisadangerousgame · Answer

Okay, that makes sense, do you mind doing another one?

dumbcow · Answer

sure

lifeisadangerousgame · Answer

$$f(x) = \frac{ x + 2 }{ x - 3 }; g(x) = \frac{ 1 }{ x + 1 }$$ I tried f(g(x)) and I got (2x + 2)/(-3x - 3) Is that right?

lifeisadangerousgame · Answer

Wait. I messed up, that's the same as the first problem

dumbcow · Answer

almost , should be (2x+3)/(-3x-2)

lifeisadangerousgame · Answer

Why's that?

dumbcow · Answer

$$\frac{1}{x+1}+\frac{2(x+1)}{x+1} = \frac{2x+3}{x+1}$$

lifeisadangerousgame · Answer

I'm still a bit lost, I'm sorry

dumbcow · Answer

hmm ok i was just combining terms into 1 fraction....

anyway you had a different question

lifeisadangerousgame · Answer

$$f(x) = \frac{ t-2 }{ t +3 }, g(x) = \frac{ 1 }{ (t+2)^2 }$$

lifeisadangerousgame · Answer

Actually, I'm still lost with how to do the first problem...I don't understand how it's )(2x + 3)/(-3x -2)

mathmale · Answer

Dumbcow did a good job of showing his work (above).  It'd be easier for potential helpers to help you if you'd show (share) any work you've done yourself.  By "first problem," do you mean your original post?