Question

Sketch the given function

f(x) = { x^2. x>=0
{ -x-2. x<0

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Answer 1

Answer:

f(x)={xifx≥0xifx<0}

to check for continuity we find the left hand limit and right hand limit. If for a point 'a' in the domain f(x)

⇒LHL=RHL=f(a)

Then the function is continuous at x=a

for x>0

f(x)=X,

and we know that f(x)=X ( identity function ) is a rational function well defined in the interval (0,∞) and is continuous,

For x<0

f(x)=x2

this is a rational function in 'x' and hence is continuous in the above domain.

However the definition of the function changes at x=0 hence we need to check for continuity at x=0

LHL : limx→0−f(x)=limh→0f(0−h)=limh→0(0−h)2=limh→0h2=0

RHL :limx→0+f(x)=limh→0f(0+h)=limh→0(0+h)=limh→0h=0

∴LHL=RHL=f(0)

Hence, the function is continuous at x=0

∴ the function is continuous x∈R i.e, is continuous through out its domain.

 Hence, solved.