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

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

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

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

Substituting $2$ for $y$

Let $2$ be substituted for $y$ in $\map f {x, y}$.

Then:

\(\ds \forall x \in \R: \, \)

\(\ds \map f {x, 2}\)

\(=\)

\(\ds x^2 + x \times 2^2 + 5 \times 2 + 3\)

\(\ds \)

\(=\)

\(\ds x^2 + 4 x + 13\)

simplifying

Substituting $a$ for $y$

Let $a$ be substituted for $y$ in $\map f {x, y}$.

Then:

\(\ds \forall x \in \R: \, \)

\(\ds \map f {x, 2}\)

\(=\)

\(\ds x^2 + x \times a^2 + 5 \times a + 3\)

\(\ds \)

\(=\)

\(\ds x^2 + a^2 x + 5 a + 3\)

simplifying

Substituting $\map g x$ for $y$

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:

\(\ds \forall x \in \R: \, \)

\(\ds \map f {x, \map g x}\)

\(=\)

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