https://file.aiquickdraw.com/mathbot/file/1727306597767vc7b9itl.jpg

From the image, the problem involves two functions:

- \( f(x) = 8x - 5 \)
- \( g(x) = -4x^2 \)

The goal is to evaluate the following operations:

a) \( f + g \)(x)  
b) \( f - g \)(x)  
c) \( f \times g \)(x)  
d) \( \frac{f}{g} \)(x)  
e) \( g \circ f \)(x)

Let’s solve each part step by step.

### a) \( f + g \)(x)
The sum of two functions is the sum of their expressions:
\[
(f + g)(x) = f(x) + g(x) = (8x - 5) + (-4x^2) = -4x^2 + 8x - 5
\]

### b) \( f - g \)(x)
The difference between the two functions is:
\[
(f - g)(x) = f(x) - g(x) = (8x - 5) - (-4x^2) = 8x - 5 + 4x^2 = 4x^2 + 8x - 5
\]

### c) \( f \times g \)(x)
Multiply the two functions:
\[
(f \times g)(x) = f(x) \cdot g(x) = (8x - 5) \cdot (-4x^2)
\]
Distribute:
\[
= 8x \cdot (-4x^2) + (-5) \cdot (-4x^2) = -32x^3 + 20x^2
\]

### d) \( \frac{f}{g} \)(x)
This is the quotient of the two functions:
\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{8x - 5}{-4x^2}
\]
This expression is already simplified.

### e) \( g \circ f \)(x)
This is the composition of \( g(x) \) with \( f(x) \), meaning we substitute \( f(x) \) into \( g(x) \):
\[
(g \circ f)(x) = g(f(x)) = g(8x - 5)
\]
Since \( g(x) = -4x^2 \), substitute \( 8x - 5 \) into \( g \):
\[
= -4(8x - 5)^2
\]
First, expand \( (8x - 5)^2 \):
\[
(8x - 5)^2 = 64x^2 - 80x + 25
\]
Now multiply by \( -4 \):
\[
= -4(64x^2 - 80x + 25) = -256x^2 + 320x - 100
\]

Would you like further clarifications on any part? Here are some related questions for further practice:

1. What is the general formula for combining functions through addition or subtraction?
2. How do you apply the distributive property when multiplying two polynomials?
3. How do you simplify complex rational functions?
4. Can you find the inverse of either \( f(x) \) or \( g(x) \)?
5. How does the composition of two functions affect the domain of the resulting function?

**Tip:** When performing operations with functions, always verify if simplifications are possible to avoid complicated expressions.

The functions f(x) and g(x) are defined as f(x) = 8x - 5 and g(x) = -4x^2. Find the following: (f + g)(x), (f - g)(x), (f × g)(x), (f / g)(x), and (g ◦ f)(x).

This lesson covers operations with functions, including addition, subtraction, multiplication, division, and composition of functions. It uses the functions f(x) = 8x - 5 and g(x) = -4x^2 to demonstrate each operation step-by-step.

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