SL 2.5—Composite and inverse functions

Compose functions in the stated order.

(f∘g)(x)=f(g(x)); composition feeds the output of one rule into the next, so order matters.

Worked example
If f(x)=2x and g(x)=x+3, f∘g=2x+6 but g∘f=2x+3.

Worked example
Why can the two compositions differ? the inner function is applied first.

Common boundary
Composition is not ordinary multiplication.

Finding and checking an inverse: for $f(x)=3x-5$, solve $y=3x-5$ for $x$, giving $f^{-1}(x)=(x+5)/3$. Then $(f\circ f^{-1})(x)=3[(x+5)/3]-5=x$ and $(f^{-1}\circ f)(x)=[(3x-5)+5]/3=x$. State any domain restriction needed to make a non-one-to-one function invertible.