https://file.aiquickdraw.com/mathbot/file/1728314421483y019o44n.jpg

The image contains a math worksheet focused on function operations, particularly involving two given functions:

- \( f(x) = 2x^2 - 4x \)
- \( g(x) = 3x - 7 \)

The problems seem to involve basic operations between these functions, such as addition, subtraction, multiplication, division, and composition of functions.

Let's go over the visible problems from the image:

### a. \( (f + g)(x) \)
This represents the addition of the two functions:
\[
(f + g)(x) = f(x) + g(x)
\]
Substituting the given functions:
\[
= (2x^2 - 4x) + (3x - 7)
= 2x^2 - x - 7
\]

### b. \( (f - g)(x) \)
This is the subtraction of \( g(x) \) from \( f(x) \):
\[
(f - g)(x) = f(x) - g(x)
\]
Substituting the functions:
\[
= (2x^2 - 4x) - (3x - 7)
= 2x^2 - x - 7
\]

### c. \( (f \cdot g)(x) \)
This is the multiplication of the two functions:
\[
(f \cdot g)(x) = f(x) \cdot g(x)
\]
\[
= (2x^2 - 4x) \cdot (3x - 7)
\]
This requires distribution (as shown partially in the image):
\[
= 6x^3 - 14x^2 - 12x^2 + 28x
\]
\[
= 6x^3 - 26x^2 + 28x
\]

### d. \( \frac{f}{g}(x) \)
This is the division of \( f(x) \) by \( g(x) \):
\[
\frac{f}{g}(x) = \frac{2x^2 - 4x}{3x - 7}
\]
This is simplified as it is.

### e. \( (f \circ g)(x) \)
This represents the composition of \( f(x) \) and \( g(x) \), meaning you substitute \( g(x) \) into \( f(x) \):
\[
(f \circ g)(x) = f(g(x))
= f(3x - 7)
= 2(3x - 7)^2 - 4(3x - 7)
\]
Expanding:
\[
= 2(9x^2 - 42x + 49) - 12x + 28
= 18x^2 - 84x + 98 - 12x + 28
= 18x^2 - 96x + 126
\]

---

Do you want further details on any specific operation or have any other questions? Here are 5 related questions that might help expand your understanding:

1. How do you simplify function compositions in general?
2. What happens when two polynomials are divided, like in part (d)?
3. How do you factor polynomials, such as the results in part (c)?
4. Can you explain how to find the domain of \( \frac{f}{g}(x) \)?
5. How would the result of \( g \circ f \) differ from \( f \circ g \)?

**Tip**: When multiplying polynomials, it's useful to use the distributive property or a method like the box method to organize the terms during expansion.

Given f(x) = 2x^2 - 4x and g(x) = 3x - 7, evaluate the following operations: (f + g)(x), (f - g)(x), (f * g)(x), (f / g)(x), and (f ∘ g)(x).

Learn how to perform function operations and composition with the given functions f(x) = 2x^2 - 4x and g(x) = 3x - 7. Includes step-by-step solutions for addition, subtraction, multiplication, division, and composition. Suitable for students in Grades 9-12.

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