f(x)=9, if x<=-4, and f(x)=ax+b, if -4<x<5, and f(x)=-9, if x>=5,Find

shadsiei 2021-09-29 Answered

\(\displaystyle{f{{\left({x}\right)}}}={9},{\quad\text{if}\quad}{x}\le-{4},{\quad\text{and}\quad}{f{{\left({x}\right)}}}={a}{x}+{b},{\quad\text{if}\quad}-{4}{<}{x}{<}{5},{\quad\text{and}\quad}{f{{\left({x}\right)}}}=-{9},{\quad\text{if}\quad}{x}\ge{5},\)
Find the constants, if the function is continuous on the entire line.

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Expert Answer

sweererlirumeX
Answered 2021-09-30 Author has 11991 answers
The function is continuous, if
\(\displaystyle{\underset{{{x}\to{a}^{+}}}{{\lim}}}{f{{\left({x}\right)}}}={\underset{{{x}\to{a}^{{-}}}}{{\lim}}}{f{{\left({x}\right)}}}={f{{\left({a}\right)}}}\)
All of the given functions are continous, since \(\displaystyle{\underset{{{1}}}{{{f}}}}{\left({x}\right)},{\underset{{{3}}}{{{f}}}}{\left({x}\right)}\) are constant function and \(\displaystyle{\underset{{{2}}}{{{f}}}}{\left({x}\right)}\) is linear function. We need to check only at point where the function might break (-4 and 5):
Atx=-4
\(\displaystyle{\underset{{{x}\to-{4}^{{-}}}}{{\lim}}}{f{{\left({x}\right)}}}={\underset{{{x}\to-{4}^{{-}}}}{{\lim}}}{9}\)
=9
\(\displaystyle{\underset{{{x}\to-{4}^{+}}}{{\lim}}}{f{{\left({x}\right)}}}={\underset{{{x}\to-{4}^{{-}}}}{{\lim}}}{a}{x}+{b}\)
=a(-4)+b
=-4a+b
f(-4)=9
\(\displaystyle{\underset{{{x}\to-{4}^{{-}}}}{{\lim}}}{f{{\left({x}\right)}}}={\underset{{{x}\to-{4}^{+}}}{{\lim}}}{f{{\left({x}\right)}}}={f{{\left(-{4}\right)}}}\)
9=-4a+b=9
-4a+b=9 (I)
At x=5
\(\displaystyle{\underset{{{x}\to{5}^{+}}}{{\lim}}}{f{{\left({x}\right)}}}={\underset{{{x}\to{5}^{{-}}}}{{\lim}}}{a}{x}+{b}\)
=5a+b
\(\displaystyle{\underset{{{x}\to{5}^{+}}}{{\lim}}}{f{{\left({x}\right)}}}={\underset{{{x}\to{5}^{+}}}{{\lim}}}-{9}\)
=-9
f(5)=-9
If the condition of continuity is satisfied,
\(\displaystyle{\underset{{{x}\to{5}^{{-}}}}{{\lim}}}{f{{\left({x}\right)}}}={\underset{{{x}\to{5}^{+}}}{{\lim}}}{f{{\left({x}\right)}}}={f{{\left({5}\right)}}}\)
5a+b=-9 (II)
Subtract (II) from (I) to get:
-9a+0=18
\(\displaystyle\to{a}=-{2}\)
Substitute a=-2 in (I)
-4(-2)+b=9
8+b=9
b=1
The final answer:
a=-2
b=1
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