For all real x, f(x)=2x-1 and g(x)=x2+2. Find fg(x), gf(x) and f2(x). Evaluate fg(2) and gf(2), and

Question

For all real x, f(x)=2x-1 and g(x)=x2+2. Find fg(x), gf(x) and f2(x). Evaluate fg(2) and gf(2), and state what the results show.

Accepted Answer

$\mathrm{fg}(x)=2x^2+3$, $\mathrm{gf}(x)=(2x-1)^2+2$, $\mathrm f^2(x)=4x-3$; both values at $2$ are $11$, but the functions differ.

Two orders of composition