f(x)=9,ifx≤−4,andf(x)=ax+b,if−4&lt;x&lt;5,andf(x)=−9,ifx≥5, Find the constants, if the function is continuous on the entire line.

Question

f(x)=9,ifx≤−4,andf(x)=ax+b,if−4&amp;lt;x&amp;lt;5,andf(x)=−9,ifx≥5, Find the constants, if the function is continuous on the entire line.

unett · Accepted Answer

The function is continuous, if limx→a+f(x)=limx→a−f(x)=f(a) All of the given functions are continous, since f1(x),f3(x) are constant function and f2(x) is linear function. We need to check only at point where the function might break (-4 and 5): At x=−4 limx→−4−f(x)=limx→−4−9 =9 limx→−4+f(x)=limx→−4−ax+b =a(−4)+b =−4a+b f(−4)=9 limx→−4−f(x)=limx→−4+f(x)=f(−4) 9=−4a+b=9 −4a+b=9 (I) At x=5 limx→5+f(x)=limx→5−ax+b =5a+b limx→5+f(x)=limx→5+−9 =−9 f(5)=−9 If the condition of continuity is satisfied, limx→5−f(x)=limx→5+f(x)=f(5) 5a+b=−9 (II) Subtract (II) from (I) to get: −9a+0=18 →a=−2 Substitute a=−2 in (I) −4(−2)+b=9 8+b=9 b=1 The final answer: a=−2 b=1

f(x)=9, if x<=-4, and f(x)=ax+b, if -4<x<5, and f(x)=-9, if x>=5,Find the constants, if the function is continuous on the entire line

Answered question

Answer & Explanation

New Questions in Algebra II