Question

Suppose g(x)=ax+b, where a and b are real constants. Find all possible pairs (a,b) such that we have g(g(x))=9x+28.

Show work or explain.

Answer 1

`g(x)=ax+b\implies g(g(x))=g(ax+b)=a(ax+b)+b=a^2x+ab+b`

You have
`a^2=9\implies a=\pm3`, and
`ab+b=28`. If
`a=3`, then
`3b+b=4b=28\implies b=7`. If
`a=-3`, then
`-3b+b=-2b=28\implies b=-14`.

So the possible pairs are
`(3,7)` and
`(-3,-14)`.