Question

Find the constants a and b such that the function is continuous on the entire real line. $ f(x) = \left\{\begin{array}{ll} {\color{red}5}, & x \leq -{\color{red}2}\\ ax + b, & -{\color{red}2} < x < {\color{red}3}\\ -{\color{red}5}, & x \geq {\color{red}3} \end{array}\right. $

Answer 1

we are given

piecewise function as

f(x) =5  when x<=-2

f(x)=ax+b  when -2<x<3

f(x)=-5 when x>=3

We are given f(x) is continuous for all values of x

so, it will also be continuous at x=-2 and x=3

Continuity at x=-2:

Value of both functions must be same at x=-2

so, we get

$f(-2)=a*-2+b=5$

$-2a+b=5$

Continuity at x=3:

Value of both functions must be same at x=3

so, we get

$f(3)=a*3+b=-5$

$3a+b=-5$

so, we get system of equation as

$-2a+b=5$

$3a+b=-5$

now, we can subtract both equations

and we get

$5a=-10$

$a=-2$

now, we can find b

$-2*-2+b=5$

$b=1$

so, we get

$a=-2$

$b=1$..............Answer