The properties of function f(x) as 1) The domain of f is (-\infty,\inft

Dillard 2021-08-21 Answered
The properties of function f(x) as
1) The domain of f is (,)
2) f(3}=7
3) f is continuous st x=2
4) f is not continuous at x=4 and
5) limx+f(x)=1
To find: The equation of function f(x) which satisfies all given properties.
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Expert Answer

stuth1
Answered 2021-08-22 Author has 97 answers

Step 1
Explanation:
Consider the function
f(x)={x(x3)+70x2+1if x415if x=4
The above function is defined over all real number therefore the domain of the function is (,),
Hence the property (1) is satisfied.
Evaluating f(3),
f(3)=3(33)+7032+1=7010=7
Hence the property (2) is satisfied.
Step 2
Evaluating limx2+f(x),
limx2+f(x)=limh0f(2+h)
Rightarrwolimx2+f(x)=limh0(2+h)(2+h3)+70(2+h)2+1
limx2+f(x)=limh0(2+h)(h1)+70(2+h)2+1
limx2+f(x)=limh0h2+h+68h2+4h+5
limx2+f(x)=685
Evaluating limx2f(x),
limx2f(x)=limh0f(2h)
limx2f(x)=limh0(2h)(2h3)+70(2+h)2+1
limx2f(x)=limh0(2h)(h1

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