Let's consider the functions $f(x) = x^2 + 1$ and $g(y) = y^2 + 2$.

Sum of $f$ and $g$:

$(f + g)(x, y) = f(x) + g(y)$ $= (x^2 + 1) + (y^2 + 2)$ $= x^2 + y^2 + 3$

Difference of $f$ and $g$:

$(f - g)(x, y) = f(x) - g(y)$ $= (x^2 + 1) - (y^2 + 2)$ $= x^2 - y^2 - 1$

Product of $f$ and $g$:

$(f \cdot g)(x, y) = f(x) \cdot g(y)$ $= (x^2 + 1) \cdot (y^2 + 2)$ $= x^2y^2 + 2x^2 + y^2 + 2$

Quotient of $f$ and $g$:

$\left( \frac{f}{g} \right)(x, y) = \frac{f(x)}{g(y)}$ $= \frac{x^2 + 1}{y^2 + 2}$

Composition of $f$ and $g$:

To find the composition $(f \circ g)(x)$ and $(g \circ f)(x)$:

• $(f \circ g)(x)$: $f(g(x)) = f(x^2 + 2)$ $= (x^2 + 2)^2 + 1$ $= x^4 + 4x^2 + 4 + 1$ $= x^4 + 4x^2 + 5$

• $(g \circ f)(x)$: $g(f(x)) = g(x^2 + 1)$ $= (x^2 + 1)^2 + 2$ $= x^4 + 2x^2 + 1 + 2$ $= x^4 + 2x^2 + 3$

Summary:

1. $(f + g)(x, y) = x^2 + y^2 + 3$
2. $(f - g)(x, y) = x^2 - y^2 - 1$
3. $(f \cdot g)(x, y) = x^2y^2 + 2x^2 + y^2 + 2$
4. $\left( \frac{f}{g} \right)(x, y) = \frac{x^2 + 1}{y^2 + 2}$
5. $(f \circ g)(x) = x^4 + 4x^2 + 5$
6. $(g \circ f)(x) = x^4 + 2x^2 + 3$

Would you like more details on any of these operations, or do you have any questions?

Related Questions:

1. How do you find the inverse of a function?
2. What is the derivative of $f(x) = x^2 + 1$ and $g(y) = y^2 + 2$?
3. How do you integrate the functions $f(x)$ and $g(y)$?
4. Can you graph the functions $f(x)$ and $g(y)$?
5. What are the critical points of $f(x)$ and $g(y)$?

Tip:

Always check if functions are one-to-one before attempting to find their inverses, as only one-to-one functions have inverses that are also functions.