Review question

# Can we compose two simple functions? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9527

## Question

Let $f(x)=x+1\qquad\text{and}\qquad g(x)=2x.$ We will, for example, write $fg$ to denote the function “perform $g$ then perform $f$” so that $fg(x)=f(g(x))=2x+1.$ If $i\geq 0$ is an integer we will, for example, write $f^i$ to denote the function which performs $f$ $i$ times, so that $f^i(x)=\underbrace{fff......f}_\text{i\,\text{times}}(x)=x+i.$

1. Show that $f^2g(x)=gf(x).$

2. Note that $gf^2g(x)=4x+4.$ Find all the other ways of combining $f$ and $g$ that result in the function $4x+4$.

3. Let $i,j,k\geq0$ be integers. Determine the function $f^igf^jgf^k(x).$

4. Let $m\geq0$ be an integer. How many different ways of combining the functions $f$ and $g$ are there that result in the function $4x+4m$?