Question

How do you express the function `y=x^2+3` as a composition y=f(g(x)) of two simpler functions y=f(u) and u=g(x)?

Answer 1

`g(x) = x^2`, `f(u) = u + 3`

Explanation:

in order to find `x^2 + 3` with a given value of `x`, you would need to first square `x` to find `x^2`, then add `3` to find `x^2+3`.

the first function, then, that you would need to carry out, would be squaring `x`.

since `g(x)` is enclosed in brackets, `g(x)` is the first function that needs to be applied.

the action done when applying the function `g(x)` is squaring `x`.
hence, the result of this, the output of `g(x)` is `x^2`.

the second function is adding `3`, and this is done to the result of the first output, `x^2`. here, this is notated as `u`, since it is the result of another function.

`u+3` will therefore give you `x^2+3`.

the notation `f(g(x))` means that the function `f(x)` is being applied to the result of the function `g(x)`.