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

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Let's work through each part of the exercise step by step.

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### **Given Functions:**
- $$ f(x) = 2x + 3 $$
- $$ g(x) = x + 2 $$
- $$ h(x) = x - 5 $$
- $$ m(x) = -3x - 1 $$

### **Question 2:**
**Given:** $$ q(x) = g(x) + h(x) $$

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

To find $$ q(x) $$, add the functions $$ g(x) $$ and $$ h(x) $$:

$$
\begin{align*}
q(x) &= g(x) + h(x) \
&= (x + 2) + (x - 5) \
&= x + x + 2 - 5 \
&= 2x - 3
\end{align*}
$$

**Answer:**
$$
q(x) = 2x - 3
$$

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#### **b. In what way is the slope of $$ q(x) $$ related to the slope of $$ g(x) $$ and the slope of $$ h(x) $$? (Include increasing or decreasing in your answer. Also identify which function is increasing faster.)**

**Slopes of the Given Functions:**
- Slope of $$ g(x) $$ ($$ m_g $$) = 1
- Slope of $$ h(x) $$ ($$ m_h $$) = 1
- Slope of $$ q(x) $$ ($$ m_q $$) = 2

**Analysis:**
- Both $$ g(x) $$ and $$ h(x) $$ have positive slopes ($$ m_g = 1 $$ and $$ m_h = 1 $$), which means both functions are **increasing**.
- The slope of $$ q(x) $$ is the sum of the slopes of $$ g(x) $$ and $$ h(x) $$: $$ m_q = m_g + m_h = 1 + 1 = 2 $$.
- Since $$ m_q = 2 $$ is greater than both $$ m_g $$ and $$ m_h $$, $$ q(x) $$ is increasing **faster** than both $$ g(x) $$ and $$ h(x) $$.

**Answer:**
The slope of $$ q(x) $$ is the sum of the slopes of $$ g(x) $$ and $$ h(x) $$. Both $$ g(x) $$ and $$ h(x) $$ have a slope of 1 and are increasing. Therefore, $$ q(x) $$ has a slope of 2 and is increasing faster than both $$ g(x) $$ and $$ h(x) $$.

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#### **c. In what way is the $$ y $$-intercept of $$ q(x) $$ related to the $$ y $$-intercepts of $$ g(x) $$ and of $$ h(x) $$?**

**Y-Intercepts of the Given Functions:**
- Y-intercept of $$ g(x) $$ = 2
- Y-intercept of $$ h(x) $$ = -5

**Analysis:**
- The $$ y $$-intercept of $$ q(x) $$ is the sum of the $$ y $$-intercepts of $$ g(x) $$ and $$ h(x) $$:
$$
\text{Y-intercept of } q(x) = 2 + (-5) = -3
$$

**Answer:**
The $$ y $$-intercept of $$ q(x) $$ is the sum of the $$ y $$-intercepts of $$ g(x) $$ and $$ h(x) $$. Specifically, $$ q(x) $$ has a $$ y $$-intercept of –3.

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

**Analysis:**
- A linear equation is generally in the form $$ y = mx + b $$, where $$ m $$ is the slope and $$ b $$ is the $$ y $$-intercept.
- $$ q(x) = 2x - 3 $$ fits this form with $$ m = 2 $$ and $$ b = -3 $$.

**Answer:**
Yes, $$ q(x) $$ is the equation of a line because it is in the standard linear form $$ y = mx + b $$, where $$ m = 2 $$ is the slope and $$ b = -3 $$ is the $$ y $$-intercept.

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a. $$ q(x) = 2x - 3 $$ b. The slope of $$ q(x) $$ is the sum of the slopes of $$ g(x) $$ and $$ h(x) $$. Both $$ g(x) $$ and $$ h(x) $$ are increasing, and $$ q(x) $$ is increasing faster. c. The $$ y $$-intercept of $$ q(x) $$ is the sum of the $$ y $$-intercepts of $$ g(x) $$ and $$ h(x) $$, which is –3. d. Yes, $$ q(x) $$ is a line because it is in the form $$ y = mx + b $$.

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