Let's work with the functions you've provided:

1. \( f(x) = x^2 - 5x + 4 \)
2. \( g(x) = x + 3 \)

We can perform several operations with these functions. Let’s compute each of them:

### 1. Sum: \((f + g)(x)\)
\[
(f + g)(x) = f(x) + g(x) = (x^2 - 5x + 4) + (x + 3)
\]
Simplify by combining like terms:
\[
(f + g)(x) = x^2 - 4x + 7
\]

### 2. Difference: \((f - g)(x)\)
\[
(f - g)(x) = f(x) - g(x) = (x^2 - 5x + 4) - (x + 3)
\]
Simplify by combining like terms:
\[
(f - g)(x) = x^2 - 6x + 1
\]

### 3. Product: \((f \cdot g)(x)\)
\[
(f \cdot g)(x) = f(x) \cdot g(x) = (x^2 - 5x + 4)(x + 3)
\]
Expand the expression:
\[
(f \cdot g)(x) = x^3 + 3x^2 - 5x^2 - 15x + 4x + 12
\]
Combine like terms:
\[
(f \cdot g)(x) = x^3 - 2x^2 - 11x + 12
\]

### 4. Quotient: \(\left(\frac{f}{g}\right)(x)\)
\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x^2 - 5x + 4}{x + 3}
\]
To simplify this, we can perform polynomial long division, but first, let's check if there's a factor of \( x + 3 \) in the numerator. In this case, there isn’t a common factor, so the expression stays as it is:
\[
\left(\frac{f}{g}\right)(x) = \frac{x^2 - 5x + 4}{x + 3}
\]

### 5. Composition: \(f(g(x))\) and \(g(f(x))\)

#### Composition \(f(g(x))\):
\[
f(g(x)) = f(x + 3)
\]
Substitute \( x + 3 \) into \( f(x) \):
\[
f(g(x)) = (x + 3)^2 - 5(x + 3) + 4
\]
Expand and simplify:
\[
f(g(x)) = (x^2 + 6x + 9) - (5x + 15) + 4 = x^2 + x - 2
\]

#### Composition \(g(f(x))\):
\[
g(f(x)) = g(x^2 - 5x + 4)
\]
Substitute \( x^2 - 5x + 4 \) into \( g(x) \):
\[
g(f(x)) = (x^2 - 5x + 4) + 3
\]
Simplify:
\[
g(f(x)) = x^2 - 5x + 7
\]

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

### Related Questions:
1. How do you factor a quadratic equation like \( x^2 - 5x + 4 \)?
2. What are the roots of the function \( f(x) = x^2 - 5x + 4 \)?
3. How do you perform polynomial long division for \( \frac{x^2 - 5x + 4}{x + 3} \)?
4. What is the domain of the function \( \left(\frac{f}{g}\right)(x) = \frac{x^2 - 5x + 4}{x + 3} \)?
5. Can you explain how to graph the functions \( f(x) \) and \( g(x) \) on the same coordinate plane?

**Tip:** When composing functions, it's important to substitute the entire expression of one function into the other, making sure to carefully distribute and combine like terms.

Explore operations with the functions f(x) = x² - 5x + 4 and g(x) = x + 3. Learn how to add, subtract, multiply, divide, and compose these functions. Suitable for high school students studying algebra and function operations.

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