This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Emmanuella, let's knock this out. For the function f(x) to be continuous at x=5 and x=12, the left-hand limit, right-hand limit, and the function value must be equal at these points. The function is given by: f(x) = 1+x & if x 5 \\ ax+b & if 5 < x < 12 \\ x+2 & if x 12 Step 1: Ensure continuity at x=5. For continuity at x=5, we must have _x 5^- f(x) = _x 5^+ f(x) = f(5). _x 5^- f(x) = _x 5^- (1+x) = 1+5 = 6 _x 5^+ f(x) = _x 5^+ (ax+b) = a(5)+b = 5a+b f(5) = 1+5 = 6 Equating the limits gives our first equation: 5a+b = 6 (Equation 1) Step 2: Ensure continuity at x=12. For continuity at x=12, we must have _x 12^- f(x) = _x 12^+ f(x) = f(12). _x 12^- f(x) = _x 12^- (ax+b) = a(12)+b = 12a+b _x 12^+ f(x) = _x 12^+ (x+2) = 12+2 = 14 f(12) = 12+2 = 14 Equating the limits gives our second equation: 12a+b = 14 (Equation 2) Step 3: Solve the system of linear equations. We have the system: 1) 5a+b = 6 2) 12a+b = 14 Subtract Equation 1 from Equation 2: (12a+b) - (5a+b) = 14 - 6 7a = 8 a = (8)/(7) Step 4: Substitute the value of a into Equation 1 to find b. 5((8)/(7)) + b = 6 (40)/(7) + b = 6 b = 6 - (40)/(7) To subtract, find a common denominator: b = (42)/(7) - (40)/(7) b = (2)/(7) The values of a and b that make the function continuous are: a = (8)/(7) b = (2)/(7) Send me the next one 📸