Simplify the ff basic function 
Let f(×)=4×-3 and g(×)=×²-2×

(F-g) (×), f(-3)

(F•g) (×), f(-2)

(F÷g)(×), f (4)

Let's simplify the expressions step-by-step, and then find the values of the functions for the given inputs.

### Given Functions
- \( f(x) = 4x - 3 \)
- \( g(x) = x^2 - 2x \)

### Simplifications

1. **(F - G)(x)**  
   This represents the function \( f(x) - g(x) \):
   \[
   (F - G)(x) = (4x - 3) - (x^2 - 2x)
   \]
   \[
   = 4x - 3 - x^2 + 2x
   \]
   \[
   = -x^2 + 6x - 3
   \]

2. **f(-3)**
   \[
   f(-3) = 4(-3) - 3
   \]
   \[
   = -12 - 3
   \]
   \[
   = -15
   \]

3. **(F • G)(x)**
   This represents the function \( f(x) \times g(x) \):
   \[
   (F • G)(x) = (4x - 3)(x^2 - 2x)
   \]
   Expanding the product:
   \[
   = 4x(x^2 - 2x) - 3(x^2 - 2x)
   \]
   \[
   = 4x^3 - 8x^2 - 3x^2 + 6x
   \]
   \[
   = 4x^3 - 11x^2 + 6x
   \]

4. **f(-2)**
   \[
   f(-2) = 4(-2) - 3
   \]
   \[
   = -8 - 3
   \]
   \[
   = -11
   \]

5. **(F ÷ G)(x)**
   This represents the function \( \frac{f(x)}{g(x)} \):
   \[
   (F ÷ G)(x) = \frac{4x - 3}{x^2 - 2x}
   \]

6. **f(4)**
   \[
   f(4) = 4(4) - 3
   \]
   \[
   = 16 - 3
   \]
   \[
   = 13
   \]

### Summary

- \( (F - G)(x) = -x^2 + 6x - 3 \)
- \( f(-3) = -15 \)
- \( (F • G)(x) = 4x^3 - 11x^2 + 6x \)
- \( f(-2) = -11 \)
- \( (F ÷ G)(x) = \frac{4x - 3}{x^2 - 2x} \)
- \( f(4) = 13 \)

### Follow-up Questions

1. How do we expand and simplify polynomial expressions?
2. What are some methods to find the roots of a quadratic equation?
3. How can we determine the domain and range of a function?
4. What is the significance of function composition in mathematics?
5. How does function division work, and what should we consider regarding the domain?
6. Can you explain how to find the vertex of a quadratic function?
7. What is the relationship between the degree of a polynomial and its graph?
8. How do transformations affect the graph of a function?

**Tip:** Always check for undefined points in division of functions, where the denominator equals zero.

Learn how to simplify and evaluate functions f(x) = 4x - 3 and g(x) = x^2 - 2x. Includes step-by-step solutions for (f - g)(x), (f • g)(x), and (f ÷ g)(x) with evaluations at specific points. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation