In each of the following cases find:

$\displaystyle f \circ g$

$\displaystyle g \circ f$

$\displaystyle f \circ f$

$\displaystyle g \circ g$

c)

$\displaystyle f(x) = \sqrt{x}$

$\displaystyle g(x) = x^2$

$\displaystyle f \circ g$For

$\displaystyle = f(g(x))$

$\displaystyle = f(x^2)$

$\displaystyle = \sqrt{x^2}$

How would you simplify the answer?

$\displaystyle g \circ f$For

$\displaystyle = g(f(x)) $

$\displaystyle = (\sqrt {x})^2$

How would you simplify the answer?

Could you please check the following to see if I did them correctly and show me where I messed up?

d)

$\displaystyle f(x) = \frac {1}{x + 1}$

$\displaystyle g(x) = \frac {x}{x -1}$

$\displaystyle g \circ f$For

$\displaystyle = g (\frac {x}{x + 1})$

$\displaystyle = \frac {(\frac{x}{x + 1})}{( \frac{x}{x + 1} - 1)}$

$\displaystyle = \frac {(\frac {x}{x + 1})}{(\frac{1}{x + 1})}$

$\displaystyle = \frac {x}{x + 1} (x + 1)$

$\displaystyle = \frac {x^2 + x}{x + 1}$

$\displaystyle g \circ g$For

$\displaystyle

= g (\frac {x}{x - 1})$

$\displaystyle = \frac {(\frac {x}{x - 1})}{(\frac {x}{x -1} - 1)}$

$\displaystyle = (\frac {x}{x - 1}) (x - 1)$

$\displaystyle = \frac {x^2 - x}{x - 1}$