Question

Suppose that the functions g and h are defined as follows.6x xxoFind the compositions gøg and hoh.Simplify your answers as much as possible.(Assume that your expressions are defined for all x in the domain of the composition. You do not have to indicate the domain.)

Answer 1

Note that :

$(f\circ f)(x)=f(f(x))$

The composition of f of f of x is the same as shown above.

From the given problem, we have :

$g(x)=x^2+1$

g(g(x)) will be the function which g(x) substitutes the value of x in g(x), this will be :

$\begin{gathered} (g\circ g)(x)=g(g(x)) \\ =(x^2+1)^2+1 \\ =(x^4+2x^2+1)+1 \\ =x^4+2x^2+2 \end{gathered}$

The answer for (g o g)(x) = x^4 + 2x^2 + 2

The next function is :

$h(x)=\frac{5}{6x}$

Same as the method we used above.

(h o h)(x) is the same as h(h(x)), and this will be :

$\begin{gathered} (h\circ h)(x)=h(h(x)) \\ =\frac{5}{6(\frac{5}{6x})} \\ =\frac{\cancel{5}}{\cancel{6}(\frac{\cancel{5}}{\cancel{6}x})} \\ =\frac{1}{\frac{1}{x}} \\ =1\times\frac{x}{1} \\ =x \end{gathered}$

Therefore, the answer for (h o h)(x) = x