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The document you uploaded contains a set of mathematical problems involving functions. The given functions are:

\[
f(x) = x^2 - 4x + 5
\]
\[
g(x) = x^2 - 1
\]
\[
h(x) = x + 1
\]

You need to determine the following operations between these functions:

1. \((f+g)(x)\)
2. \((g+h)(x)\)
3. \((g-h)(x)\)
4. \((f \cdot h)(x)\)
5. \(g(h(x))\) (composition of \(g\) and \(h\))
6. \((h \cdot g)(x)\)
7. \((f-g)(x)\)
8. \(f(h(x))\) (composition of \(f\) and \(h\))

### Let's solve each one step by step:

1. **\((f+g)(x)\)**:
   \[
   (f+g)(x) = f(x) + g(x) = (x^2 - 4x + 5) + (x^2 - 1) = 2x^2 - 4x + 4
   \]

2. **\((g+h)(x)\)**:
   \[
   (g+h)(x) = g(x) + h(x) = (x^2 - 1) + (x + 1) = x^2 + x
   \]

3. **\((g-h)(x)\)**:
   \[
   (g-h)(x) = g(x) - h(x) = (x^2 - 1) - (x + 1) = x^2 - x - 2
   \]

4. **\((f \cdot h)(x)\)**:
   \[
   (f \cdot h)(x) = f(x) \cdot h(x) = (x^2 - 4x + 5) \cdot (x + 1) = x^3 - 3x^2 + x + 5
   \]

5. **\(g(h(x))\)**:
   \[
   g(h(x)) = g(x + 1) = (x + 1)^2 - 1 = x^2 + 2x
   \]

6. **\((h \cdot g)(x)\)**:
   \[
   (h \cdot g)(x) = h(x) \cdot g(x) = (x + 1) \cdot (x^2 - 1) = x^3 + x^2 - x - 1
   \]

7. **\((f-g)(x)\)**:
   \[
   (f-g)(x) = f(x) - g(x) = (x^2 - 4x + 5) - (x^2 - 1) = -4x + 6
   \]

8. **\(f(h(x))\)**:
   \[
   f(h(x)) = f(x + 1) = (x + 1)^2 - 4(x + 1) + 5 = x^2 - 2x + 2
   \]

Would you like further details on any of these calculations? 

Here are some related questions you might find interesting:

1. What is the difference between function composition and multiplication of functions?
2. How do you find the inverse of a function?
3. What happens when you compose a function with its inverse?
4. Can two different functions have the same derivative?
5. How do you determine if a function is one-to-one?

**Tip:** Always verify the domain of the functions involved when performing operations, as combining functions can sometimes result in undefined expressions.

Given the functions f(x) = x^2 - 4x + 5, g(x) = x^2 - 1, and h(x) = x + 1, determine the following operations: (f+g)(x), (g+h)(x), (g-h)(x), (f∙h)(x), g(h(x)), (h∙g)(x), (f-g)(x), and f(h(x)).

Explore detailed solutions for determining the operations and compositions of the functions f(x) = x^2 - 4x + 5, g(x) = x^2 - 1, and h(x) = x + 1. This guide is perfect for high school students learning advanced algebra.

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