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Let \(f(x) = \left\{ \begin{array}{cl} ax+3, &\text{ if }x>2, \\ x-5 &\text{ if } -2 \le x \le 2, \\ 2x-b &\text{ if } x <-2. \end{array} \right.\) Find a+b if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper).

 Aug 14, 2023
 #1
avatar+128826 
+1

Put 2 into  each of the  first two functions  for x and set them  equal

 

a(2) + 3 = 2 - 5

2a = 2 - 5 - 3

2a = -6

a = -3

 

Put - 2  into  each of the last two  functions for x and  set them equal

 

-2 - 5 = 2(-2) - b

-7 = -4 - b

b = -4 + 7

b = 3

 

a + b =  3 + -3  =   0

 

cool cool cool

 Aug 14, 2023
 #2
avatar+183 
+3

Rigorously, continuity is defined below:
 

For every \(k\) in the domain of \(f\);

  • \(f(k)\) exists
  •  \(\lim_{i\to k}f(i)\) exists (i.e. \(\lim_{i\to k^-}f(i) = \lim_{i \to k+}f(i)\))
  • and $\lim_{i\to k}f(i) = f(k)$
 Aug 14, 2023

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