Please help ### Continuity of Piecewise Function**Problem Statement:**The function, \( f(x) \), defined below, is a continuous function. Find the values of \( a \) and \( b \) that make the function continuous.\[ f(x) = \begin{cases} a \cdot x + b, & x < 2 \\3x + 5, & 2 \le x < 7 \\b \cdot x - a, & x \ge 7 \end{cases}\]**Solution Approach:**To ensure that the function \( f(x) \) is continuous across its entire domain, the left-hand limit should be equal to the right-hand limit at the points \( x = 2 \) and \( x = 7 \), and the function value at those points should match these limits.1. **Continuity at \( x = 2 \):** - The limits as \( x \) approaches 2 from the left and right must equal \( f(2) \). \( \lim_{{x \to 2^-}} f(x) = a \cdot 2 + b \) \( \lim_{{x \to 2^+}} f(x) = 3 \cdot 2 + 5 = 6 + 5 = 11 \) For continuity: \[ a \cdot 2 + b = 11 \] \[ 2a + b = 11 \quad \text{(Equation 1)} \]2. **Continuity at \( x = 7 \):** - The limits as \( x \) approaches 7 from the left and right must equal \( f(7) \). \( \lim_{{x \to 7^-}} f(x) = 3 \cdot 7 + 5 = 21 + 5 = 26 \) \( \lim_{{x \to 7^+}} f(x) = b \cdot 7 - a \) For continuity: \[ 7b - a = 26 \quad \text{(Equation 2)} \]**Solving the Equations:**We have the system of equations:1. \( 2a + b = 11 \) 2. \( 7b -

Name: Please help ### Continuity of Piecewise Function**Problem Statement:**
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: Please help ### Continuity of Piecewise Function**Problem Statement:**The function, \( f(x) \), defined below, is a continuous function. Find the values of \( a \) and \( b \) that make the function continuous.\[ f(x) = \begin{cases} a \cdot x + b, &