If $f(x)$ and $g(x)$ are two functions with $g(x)=x-\frac{1}{x}$ and $f \circ g(x)=x^3-\frac{1}{x^3}$,then $f^{\prime}(x)$ is equal to

If $f(x) = \frac{\alpha x}{x + 1}$,$x \ne -1$,for what value of $\alpha$ is $f(f(x)) = x$?

Let $f(x) = \sin x$ and $g(x) = \cos x$. Which of the following statements is false?

Let $f$ and $g$ be two functions defined by $f(x) = \begin{cases} x+1, & x < 0 \\ |x-1|, & x \geq 0 \end{cases}$ and $g(x) = \begin{cases} x+1, & x < 0 \\ 1, & x \geq 0 \end{cases}$. Then $(g \circ f)(x)$ is

Show that if $f: A \rightarrow B$ and $g: B \rightarrow C$ are onto,then $g \circ f: A \rightarrow C$ is also onto.

Give examples of two functions $f: N \rightarrow Z$ and $g: Z \rightarrow Z$ such that $g \circ f$ is injective but $g$ is not injective. (Hint: Consider $f(x) = x$ and $g(x) = |x|$)