Question

Find gof if
f(x)=x+(1/x) and g(x)= (x+1)/(x+2)

Answer 1

g(f(x))= (f(x)+1)/(f(x)+2)

Answer 2

just out the value of f(x) in place of f(x) in above equation n solve

Answer 3

@Theanitrix

Answer 4

put*

Answer 5

I know that, I just am not sure how to solve it

Answer 6

ok

Answer 7

the fraction confuses me

Answer 8

\[(g \circ f) = g\left(x+\frac{1}{x}\right) = \frac{\left(x+\dfrac{1}{x}\right)+1}{\left(x+\dfrac{1}{x}\right)+2}\]

Answer 9

well, you got the answer now, I guess.

Answer 10

That isn't the answer.

Answer 11

that's almost the final answer @Jhannybean

Answer 12

the answer would be (x^2+x+1)/(x+1)^2

Answer 13

but i want to know how they got that answer

Answer 14

multiply both the numerator and denominator by \(x\). \[\frac{\left(x+\dfrac{1}{x}\right)+1}{\left(x+\dfrac{1}{x}\right)+2} \cdot \frac{x}{x} \]

Answer 15

(g∘f)=g(x+1/x)=(x^2+1+x)/(x^2+2+x)

Answer 16

is that because it is the denominator of 1/x? @Jhannybean

Answer 17

Yes @Theanitrix

Answer 18

okay so would that same thing apply to fof?

Answer 19

\[(f\circ f)= f\left(x+\frac{1}{x}\right) = \left(x+\frac{1}{x}\right)+\dfrac{1}{x+\dfrac{1}{x}}\]

Answer 20

but when you're trying to solve this... set all that complicated junk = \(a\) or any other variable. find the common denominator, and simplify.

Answer 21

\[\text{let a}=x+\frac{1}{x}\]\[(f\circ f) = a+\frac{1}{a} = \frac{a^2+1}{a} = \dfrac{\left(x+\dfrac{1}{x}\right)^2+1}{\left(x+\dfrac{1}{x}\right)} \]

Answer 22

\[=\frac{x^2+\dfrac{1}{x^2}+2 +1}{\dfrac{x^2+1}{x}} =\frac{x^2+\dfrac{1}{x^2}+3}{\dfrac{x^2+1}{x}} \] So now you'd want to multiply both numerator and denominator by \(\dfrac{x}{x}\)

Answer 23

i did it a different way

Answer 24

and got the answer x^4+3x^2+1/x(x^2+1)

Answer 25

Thats correct

Answer 26

okay, i understood your approach as well so thank you

Answer 27

\[\frac{x^2+\dfrac{1}{x^2}+3}{\dfrac{x^2+1}{x}} \cdot \frac{x}{x} =\frac{\dfrac{x^4+3x^2+1}{x}}{x^2+1} = \color{red}{\frac{x^4+3x^2+1}{x(x^2+1)}}\]