In the following exercise, you will be given the opportunity to recall what happens when you add two linear functions. Given: $$ f(x)=2 x+3, g(x)=x+2 $$, $$ h(x)=x-5, m(x)=-3 x-1 $$ 1. $$ p(x)=f(x)+g(x) $$ a. Write the equation of $$ p(x) $$. b. In what way is the slope of $$ p(x) $$ related to the slope of $$ f(x) $$ and the slope of $$ g(x) $$ ? (Include increasing or decreasing in your answer. Also identify which function is increasing faster.) c. In what way is the $$ y $$-intercept of $$ p(x) $$ related to the $$ y $$-intercepts of $$ f(x) $$ and of $$ g(x) $$ ? d. Is $$ p(x) $$ the equation of a line? Justify your answer.

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Certainly! Let's address each part of the exercise step-by-step.

### **1. $$ p(x) = f(x) + g(x) $$**

#### **a. Write the equation of $$ p(x) $$.**

To find $$ p(x) $$, simply add the corresponding expressions of $$ f(x) $$ and $$ g(x) $$:

$$
\begin{align*}
p(x) &= f(x) + g(x) \
&= (2x + 3) + (x + 2) \
&= 2x + x + 3 + 2 \
&= 3x + 5
\end{align*}
$$

**Answer:**  
$$ p(x) = 3x + 5 $$

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#### **b. How is the slope of $$ p(x) $$ related to the slopes of $$ f(x) $$ and $$ g(x) $$? Include increasing or decreasing in your answer. Also, identify which function is increasing faster.**

- **Slopes of the Original Functions:**
  - $$ f(x) = 2x + 3 $$ has a slope of **2**.
  - $$ g(x) = x + 2 $$ has a slope of **1**.

- **Slope of $$ p(x) $$:**
  - $$ p(x) = 3x + 5 $$ has a slope of **3**.

- **Relationship:**
  - The slope of $$ p(x) $$ is the **sum** of the slopes of $$ f(x) $$ and $$ g(x) $$.
  - Since all slopes are positive, all functions are **increasing**.
  
- **Which Function is Increasing Faster:**
  - $$ f(x) $$ has a slope of 2, which is **steeper** than $$ g(x) $$ with a slope of 1.
  - Therefore, $$ f(x) $$ is **increasing faster** than $$ g(x) $$.

**Answer:**  
The slope of $$ p(x) $$ is the sum of the slopes of $$ f(x) $$ and $$ g(x) $$. Since both $$ f(x) $$ and $$ g(x) $$ have positive slopes, $$ p(x) $$ is also increasing. Additionally, $$ f(x) $$ (slope 2) is increasing faster than $$ g(x) $$ (slope 1).

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#### **c. How is the $$ y $$-intercept of $$ p(x) $$ related to the $$ y $$-intercepts of $$ f(x) $$ and $$ g(x) $$?**

- **Y-Intercepts of the Original Functions:**
  - $$ f(x) = 2x + 3 $$ has a $$ y $$-intercept of **3**.
  - $$ g(x) = x + 2 $$ has a $$ y $$-intercept of **2**.

- **Y-Intercept of $$ p(x) $$:**
  - $$ p(x) = 3x + 5 $$ has a $$ y $$-intercept of **5**.

- **Relationship:**
  - The $$ y $$-intercept of $$ p(x) $$ is the **sum** of the $$ y $$-intercepts of $$ f(x) $$ and $$ g(x) $$.
  
**Answer:**  
The $$ y $$-intercept of $$ p(x) $$ is the sum of the $$ y $$-intercepts of $$ f(x) $$ and $$ g(x) $$. Specifically, $$ 3 + 2 = 5 $$.

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#### **d. Is $$ p(x) $$ the equation of a line? Justify your answer.**

Yes, $$ p(x) $$ is the equation of a line.

- **Reasoning:**
  - A linear function has the form $$ y = mx + b $$, where $$ m $$ is the slope and $$ b $$ is the $$ y $$-intercept.
  - $$ p(x) = 3x + 5 $$ fits this form with a slope of 3 and a $$ y $$-intercept of 5.
  
**Answer:**  
Yes, $$ p(x) $$ is the equation of a line because it is in the form $$ y = mx + b $$ with a constant slope and $$ y $$-intercept.
**1. $$ p(x) = f(x) + g(x) $$** a. $$ p(x) = 3x + 5 $$ b. The slope of $$ p(x) $$ is the sum of the slopes of $$ f(x) $$ and $$ g(x) $$. Both functions are increasing, and $$ f(x) $$ is increasing faster. c. The $$ y $$-intercept of $$ p(x) $$ is the sum of the $$ y $$-intercepts of $$ f(x) $$ and $$ g(x) $$. d. Yes, $$ p(x) $$ is a line because it is in the form $$ y = mx + b $$.

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