Need Help with some questions please.

[susu]

I need help with one question. I solved some of it but i'm not getting the other part.

1. Find the functions f o g, g o f, f o f, and g o g and their domains.
i know f o g = f(x+5)/x and it's domain is ( negative infinity to -5) U (-5,0) U (0 to infinity)
But i don't know what g o f would be?

I know f o f is x but i'm confused on what the domain would be and i'm also confused on what g o g would be . ?

2.Which of the given functions are one- to -one?
a) h(X)= x^4+5
b)g(x)=6 squareroot x
c)f(x)= 1/x
d) g(x)= 6x^2-6x
e)f(x)= -6x+6
I thought it would be only a , b , and c but i'm still kind of unsure.



PLEASE AND THANK YOU

 

[tkhunny]

1. Don't we need to know f(x) and g(x) first?

 

[susu]

Oh i apologize!
Okay, f(x)= 5/x and g(x)=x/x+5

 

[soroban]

Hello, susu!

Given: .\(\displaystyle f(x) \,=\,\dfrac{5}{x},\;\;g(x) \,=\,\dfrac{x}{x+5}\)

1. Find the functions: .\(\displaystyle f\!\circ\!g,\; g\!\circ\!f,\; f\!\circ\!f\), and \(\displaystyle g\!\circ\!g\), and their domains.

\(\displaystyle f\!\circ\!g \;=\;f(g(x)) \;=\;f\left(\frac{x}{x+5}\right) \;=\;\dfrac{5}{\frac{x}{x+5}} \;=\;\dfrac{5(x+5)}{x} \)

. . Domain: \(\displaystyle x\ne0\qquad (\text{-}\infty,0) \cup (0, \infty)\)



\(\displaystyle g\!\circ\!f \;=\;g(f(x)) \;=\;g\left(\frac{5}{x}\right) \;=\;\dfrac{\frac{5}{x}}{\frac{5}{x} + 5}\;=\;\dfrac{5}{5+5x} \;=\;\dfrac{1}{1+x} \)

. . Domain: .\(\displaystyle x \ne \text{-}1 \qquad (\text{-}\infty,\text{-}1) \cup (\text{-}1, \infty)\)



\(\displaystyle f\!\circ\!f \;=\;f(f(x)) \;=\;f\left(\frac{5}{x}\right) \;=\;\dfrac{5}{\frac{5}{x}} \;=\;x\)

. . Domain: .all real \(\displaystyle x\qquad (\text{-}\infty, \infty)\)



\(\displaystyle g\!\circ\!g \;=\;g(g(x)) \;=\;g\left(\frac{x}{x+5}\right) \;=\;\dfrac{\frac{x}{x+5}}{\frac{x}{x+5} + 5} \;=\;\dfrac{x}{x+5(x+5)} \;=\;\dfrac{x}{6x+25} \)

. . Domain: .\(\displaystyle x \ne \text{-}\frac{25}{6} \qquad \left(\text{-}\infty, \text{-}\frac{25}{6}\right) \cup \left(\text{-}\frac{26}{6},\infty\right)\)

 

[susu]

After doing it several times i figured it out! But your explanation was great! Thank you sooooo much! Aprreciate it!

 

[lookagain]

soroban and susu,

your domains are not worked out correctly. The domain of the input function
must also be considered when determining the overall domain of the composite.


First one:

\(\displaystyle Also, \ x \ne -5.\)



Second one:

\(\displaystyle Also, \ x \ne \ 0.\)



Third one:

\(\displaystyle x \ne \ 0.\)



Fourth one:

\(\displaystyle Also, \ x \ne \ -5\)




One of many sources:

http://www.sinclair.edu/centers/mat...116,117/CompositeFunctionsAndTheirDomains.pdf