Real Function of Two Variables/Substitution for y/Examples/x^2 + xy^2 + 5y + 3/g (x)

Example of Substitution for $y$ in Real Function of Two Variables

Let $\map f {x, y}$ be the real function of $2$ variables defined as:

$\forall \tuple {x, y} \in \R^2: \map f {x, y} := x^2 + x y^2 + 5 y + 3$

Let $\map g x$ be substituted for $y$ in $\map f {x, y}$, where $\map g x$ is a real function defined on all $\R$.

Then:

$\ds \forall x \in \R: \, $

$\ds \map f {x, \map g x}$

$=$

$\ds x^2 + x \times \paren {\map g x}^2 + 5 \map g x + 3$

1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text C$: Function of Two Independent Variables: Example $2.66 \ \text {(c)}$