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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. $

Respuesta :

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

[tex]f(-2)=a*-2+b=5[/tex]

[tex]-2a+b=5[/tex]

Continuity at x=3:

Value of both functions must be same at x=3

so, we get

[tex]f(3)=a*3+b=-5[/tex]

[tex]3a+b=-5[/tex]

so, we get system of equation as

[tex]-2a+b=5[/tex]

[tex]3a+b=-5[/tex]

now, we can subtract both equations

and we get

[tex]5a=-10[/tex]

[tex]a=-2[/tex]

now, we can find b

[tex]-2*-2+b=5[/tex]

[tex]b=1[/tex]

so, we get

[tex]a=-2[/tex]

[tex]b=1[/tex]..............Answer

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