Precalculus · Problem 5.1.40

The key is to recognize that this rational function naturally breaks into an "outer" function that operates on a simpler "inner" expression. Notice that the denominator $x^4 + 1$ can be rewritten in terms of $x^2$, so if you let the inner function produce $x^2$, the outer function can then work with that squared term. This transforms a complicated-looking rational expression into a clean composition—just pick your inner and outer functions strategically, then verify by substituting back.

The key is to recognize that this rational function naturally breaks into an "outer" function that operates on a simpler "inner" expression. Notice that the denominator $x^4 + 1$ can be rewritten in terms of $x^2$, so if you let the inner function produce $x^2$, the outer function can then work with that squared term. This transforms a complicated-looking rational expression into a clean composition—just pick your inner and outer functions strategically, then verify by substituting back.