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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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.

stuth1 · Accepted Answer

Step 1Explanation:Consider the functionf(x)={x(x−3)+70x2+1if&amp;nbsp;x≠415if&amp;nbsp;x=4&amp;nbsp;The above function is defined over all real number therefore the domain of the function is&amp;nbsp;(−∞,∞),Hence the property (1) is satisfied.Evaluating&amp;nbsp;f(3),f(3)=3(3−3)+7032+1=7010=7Hence the property (2) is satisfied.Step 2Evaluating&amp;nbsp;limx→2+f(x),limx→2+f(x)=limh→0f(2+h)⇒limx→2+f(x)=limh→0(2+h)(2+h−3)+70(2+h)2+1⇒limx→2+f(x)=limh→0(2+h)(h−1)+70(2+h)2+1⇒limx→2+f(x)=limh→0h2+h+68h2+4h+5⇒limx→2+f(x)=685Evaluating&amp;nbsp;limx→2−f(x),limx→2−f(x)=limh→0f(2−h)⇒limx−2−f(x)=limh→0(2−h)(2−h−3)+70(2+h)2+1⇒limx−2−f(x)=limh→0

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

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