Question

Given that F(x) = cos2(x + 9), find functions f,g and h such that F = fogoh

Answer 1

The functions
`f(x)`,
`g(x)` and
`h(x)` such that
`F(x) = f\,\circ\,g\,\circ \,h (x)` are
`f(x) = x^(2)`,
`g(x) = \cos x` and
`h(x) = x+9`, respectively.

Explanation:

Let
`F(x) = \cos ^(2)(x+9)`, if
`F(x) = f\,\circ\,g\,\circ \,h (x)`, then we derive
`f(x)`,
`g(x)` and
`h(x)` by using the following approach:

1)
`F(x) = \cos ^(2) (x+9)` Given

2)
`f\,\circ\,g(x) = \cos^(2) x` Definition of function composition/
`h(x) = x+9`

3)
`f(x) = x^(2)` Definition of function composition/
`g(x) = \cos x`

4)
`f(x) = x^(2)`,
`g(x) = \cos x`,
`h(x) = x+9` From 2) and 3)/Result

The functions
`f(x)`,
`g(x)` and
`h(x)` such that
`F(x) = f\,\circ\,g\,\circ \,h (x)` are
`f(x) = x^(2)`,
`g(x) = \cos x` and
`h(x) = x+9`, respectively.