Given the following functions, find each:

f
(
x
)
=
x
2
−
2
x
−
8
 

g
(
x
)
=
x
−
4
 

(
f
+
g
)
(
x
)
=
 
  

(
f
−
g
)
(
x
)
=
 
  

(
f
⋅
g
)
(
x
)
=
 
  

(
f
g
)
(
x
)
=

Let's start by defining the given functions:

1. \( f(x) = x^2 - 2x - 8 \)
2. \( g(x) = x - 4 \)

### For \( (f + g)(x) \):

\[
(f + g)(x) = f(x) + g(x) = (x^2 - 2x - 8) + (x - 4)
\]

Simplifying this:

\[
(f + g)(x) = x^2 - 2x - 8 + x - 4
\]

\[
(f + g)(x) = x^2 - x - 12
\]

### For \( (f - g)(x) \):

\[
(f - g)(x) = f(x) - g(x) = (x^2 - 2x - 8) - (x - 4)
\]

Simplifying this:

\[
(f - g)(x) = x^2 - 2x - 8 - x + 4
\]

\[
(f - g)(x) = x^2 - 3x - 4
\]

### For \( (f \cdot g)(x) \):

\[
(f \cdot g)(x) = f(x) \cdot g(x) = (x^2 - 2x - 8)(x - 4)
\]

We'll expand this expression:

\[
(x^2 - 2x - 8)(x - 4) = x^2(x - 4) - 2x(x - 4) - 8(x - 4)
\]

Expanding each term:

\[
= x^3 - 4x^2 - 2x^2 + 8x - 8x + 32
\]

Simplifying:

\[
(f \cdot g)(x) = x^3 - 6x^2 + 32
\]

### For \( (f / g)(x) \):

\[
(f / g)(x) = \frac{f(x)}{g(x)} = \frac{x^2 - 2x - 8}{x - 4}
\]

We can attempt to factor the numerator and simplify. Factor \( x^2 - 2x - 8 \):

\[
x^2 - 2x - 8 = (x - 4)(x + 2)
\]

Thus, we have:

\[
(f / g)(x) = \frac{(x - 4)(x + 2)}{x - 4}
\]

As long as \( x \neq 4 \) (to avoid division by zero), we can cancel \( (x - 4) \) from both the numerator and the denominator:

\[
(f / g)(x) = x + 2 \quad \text{for} \quad x \neq 4
\]

### Summary of results:

1. \( (f + g)(x) = x^2 - x - 12 \)
2. \( (f - g)(x) = x^2 - 3x - 4 \)
3. \( (f \cdot g)(x) = x^3 - 6x^2 + 32 \)
4. \( (f / g)(x) = x + 2 \) for \( x \neq 4 \)

Would you like further details or explanations on any part of the solution?

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**Related Questions**:
1. How do you factor quadratic equations like \( x^2 - 2x - 8 \)?
2. What is the difference between polynomial and rational functions?
3. How do you simplify rational expressions with common factors?
4. What happens when you divide by zero in a rational function?
5. How do you expand polynomials in multiplication?

**Tip**: When simplifying expressions involving polynomials, always try factoring first, especially if there's a common factor you can cancel.

Given the following functions, find each:

f(x) = x^2 - 2x - 8
g(x) = x - 4
(f + g)(x) = 
(f - g)(x) = 
(f ⋅ g)(x) = 
(f / g)(x) = 

Learn how to perform operations like addition, subtraction, multiplication, and division on functions with this step-by-step solution. This example covers the functions f(x) = x^2 - 2x - 8 and g(x) = x - 4, and how to simplify the resulting expressions. Suitable for students in Grades 9-11.

Math Problem Statement

f ( x )

g ( x )

( f + g ) ( x )

( f − g ) ( x )

( f ⋅ g ) ( x )

( f g ) ( x )

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