f (x)=3x +3                   g(x)= -4x + 1

Given the functions \( f(x) = 3x + 3 \) and \( g(x) = -4x + 1 \), let's explore various operations and properties related to these functions.

1. **Sum of the Functions:**

\[ (f + g)(x) = f(x) + g(x) \]
\[ (f + g)(x) = (3x + 3) + (-4x + 1) \]
\[ (f + g)(x) = 3x + 3 - 4x + 1 \]
\[ (f + g)(x) = -x + 4 \]

2. **Difference of the Functions:**

\[ (f - g)(x) = f(x) - g(x) \]
\[ (f - g)(x) = (3x + 3) - (-4x + 1) \]
\[ (f - g)(x) = 3x + 3 + 4x - 1 \]
\[ (f - g)(x) = 7x + 2 \]

3. **Product of the Functions:**

\[ (f \cdot g)(x) = f(x) \cdot g(x) \]
\[ (f \cdot g)(x) = (3x + 3)(-4x + 1) \]
\[ (f \cdot g)(x) = 3x(-4x) + 3x(1) + 3(-4x) + 3(1) \]
\[ (f \cdot g)(x) = -12x^2 + 3x - 12x + 3 \]
\[ (f \cdot g)(x) = -12x^2 - 9x + 3 \]

4. **Quotient of the Functions:**

\[ \left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} \]
\[ \left( \frac{f}{g} \right)(x) = \frac{3x + 3}{-4x + 1} \]

This function is undefined for \( x \) values that make the denominator zero. Let's find the values of \( x \) for which \( -4x + 1 = 0 \):

\[ -4x + 1 = 0 \]
\[ -4x = -1 \]
\[ x = \frac{1}{4} \]

So, the quotient \( \left( \frac{f}{g} \right)(x) \) is undefined at \( x = \frac{1}{4} \).

5. **Composition of the Functions:**

- \( (f \circ g)(x) = f(g(x)) \)

\[ (f \circ g)(x) = f(-4x + 1) \]
\[ f(-4x + 1) = 3(-4x + 1) + 3 \]
\[ f(-4x + 1) = -12x + 3 + 3 \]
\[ f(-4x + 1) = -12x + 6 \]

- \( (g \circ f)(x) = g(f(x)) \)

\[ (g \circ f)(x) = g(3x + 3) \]
\[ g(3x + 3) = -4(3x + 3) + 1 \]
\[ g(3x + 3) = -12x - 12 + 1 \]
\[ g(3x + 3) = -12x - 11 \]

6. **Intersection Points:**

To find the points where \( f(x) = g(x) \):

\[ 3x + 3 = -4x + 1 \]
\[ 3x + 4x = 1 - 3 \]
\[ 7x = -2 \]
\[ x = -\frac{2}{7} \]

Substitute \( x = -\frac{2}{7} \) into \( f(x) \) or \( g(x) \) to find the \( y \)-coordinate:

\[ f\left( -\frac{2}{7} \right) = 3\left( -\frac{2}{7} \right) + 3 \]
\[ f\left( -\frac{2}{7} \right) = -\frac{6}{7} + 3 \]
\[ f\left( -\frac{2}{7} \right) = -\frac{6}{7} + \frac{21}{7} \]
\[ f\left( -\frac{2}{7} \right) = \frac{15}{7} \]

So, the intersection point is \( \left( -\frac{2}{7}, \frac{15}{7} \right) \).

Would you like more details on any specific operation or have any questions? 

Here are some related questions to explore:

1. What is the domain and range of \( f(x) \) and \( g(x) \)?
2. How do you graph \( f(x) \) and \( g(x) \)?
3. What are the roots of \( f(x) \) and \( g(x) \)?
4. How do the slopes of \( f(x) \) and \( g(x) \) compare?
5. What are the y-intercepts of \( f(x) \) and \( g(x) \)?
6. How can we find the inverse of \( f(x) \) and \( g(x) \)?
7. What are the applications of linear functions like \( f(x) \) and \( g(x) \) in real-life problems?
8. How do you determine if two functions are perpendicular or parallel based on their equations?

**Tip:** When combining functions, always perform the operations carefully and simplify the expressions step by step to avoid errors.

Explore operations on linear functions f(x) = 3x + 3 and g(x) = -4x + 1, including addition, subtraction, multiplication, division, composition, and finding intersection points. Detailed solutions and explanations provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation