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.

Let's work through each part of the exercise step by step. --- ### **Given Functions:** - \( f(x) = 2x + 3 \) - \( g(x) = x + 2 \) - \( h(x) = x - 5 \) - \( m(x) = -3x - 1 \) ### **Question 2:** **Given:** \( q(x) = g(x) + h(x) \) --- #### **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:** \[ q(x) = 2x - 3 \] --- #### **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 \( h(x) \). --- #### **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. --- #### **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. ---