Question

How do you find the compositions given `f(x) = x+4/3` and `g(x) = 3x-4`?

Answer 1

`{((f@g)(x)=3x-8/3),((g@f)(x)=3x):}`

Explanation:

Composition functions are when you take one function and plug the entire function into another function.

For example, if we wanted to calculate `(f@g)(x)`, we would take `g(x)`, which is `3x-4`, and plug it in for the `x` in `f(x)`.

Let's try:

If `f(x)=x+4/3`:
`(f@g)(x)=overbrace((3x-4))^("x replaced by g(x)")+4/3`

We can simplify this by adding `-4` and `4/3`.

`(f@g)(x)=3x-8/3`

We can do the same process, just switching which function is plugged into which, to find `(g@f)(x)`.

If `g(x)=3x-4`:
`(g@f)(x)=3overbrace((x+4/3))^("x replaced by f(x)")-4`

Distribute the `3`.

`(g@f)(x)=3x+4-4`

`(g@f)(x)=3x`