f=x^2+1
g=y^2+2

f+g, f-g,fxg,f/g, f o g solve

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.

f=x^2+1, g=y^2+2; f+g, f-g, f x g, f/g, f o g solve

Explore how to solve various operations on functions such as addition, subtraction, multiplication, division, and composition using the example functions f(x) = x^2 + 1 and g(y) = y^2 + 2. This guide covers detailed step-by-step solutions and explanations, suitable for high school students.

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