Question

For what values of a and b is

g(x)= { ax +2b, x is less than and equal to -2 }
{ x squared + 3a- b, -2< x less than and equal to 3 }
{3x-5, x> 3}

Answer 1

In order for the function to be continuous it is necessary that the pieces of function $g(x)$ are equal in points of discontinuity.

The points of discontinuity are the x = -2 and x = 3. Then:

$\begin{cases} ax + 2b = x^2 +3a - b, x= -2 \\ x^2 +3a - b = 3x - 5, x = 3 \end{cases}$ $\implies$ $\begin{cases} -2a + 2b = 4 + 3a - b, \\ 9 +3a - b = 9 - 5 \end{cases}$ $\implies$ $\begin{cases} 3b = 4 + 5a, \\ 3a - b = -5 \end{cases}$$\implies$

$\begin{cases} b = \frac{4 + 5a}{3} , \\ b = 3a + 5 \end{cases}$ $\implies$ $\frac{4 + 5a}{3} = 3a + 5$

$4 + 5a = 9a + 15 \implies a = -\frac{11}{4}$ $\implies b = 3* (-\frac{11}{4}) + 5 = -\frac{13}{4}$

Therefore, $g(x)$ is continuous when $a = -\frac{11}{4} , b = -\frac{13}{4}$