Math

Here we need to use 3+. which means x is little more than 3. Example 3.000001 or 3.001 or 3.02 or a number close to 3 but little larger than 3 Therefore, if we plug in x= 3.000001 in the function, we get  and if we keep reducing the value of x but ensuring it is still larger than 3, we can observe that the final answer keeps increasing and getting really large or  Examples of such values of x are:3.000000000000000000005 or 3.0000000000000000000001 You can plug in the above values and verify the result to be really large. and hence the Case2: Here it is 3-, which means x is close to 3 but little less than 3 Try to think of such numbers! Anyways, Examples of x are 2.99999 or 2.99998 and infinitely more. Lets plug in one of such values of x in the expression which is close to 3 but less than 3, Usin x=2.99999 in the expression and if we keep increasing the value of x but ensuring it is still less than 3, we can observe that the final answer keeps decreasing and getting really sma...

 f(x) =-x^3 +x^2 Step 1: Graphing. |f(x)| from f(x) So, reflecting the part of f(x) below X-Axis to above it Step2: f(|x|) from |f(x)| Getting rid of |f(x)| on the left side of the X-axis  Finally f(|x|) from |f(x)|  By Reflecting the right side of |f(x)| on the left side of X-axis i.e |f(|x|) graph

Graphing |f(|x|)| from f(x) can be broken into two steps f(x)=sinx Step1 : f(|x|) from f(x) Getting rid of the f(x) on negative X-Axis and f(x) = f(|x|) on the positive or right side of X-Axis. Reflecting f(|x|) on the left side f(|x|) Graph is below Step2: Sketching |f(x)| using f(|x|) i.e. reflecting the part of f(|x|) that is below the X-Axis , above the X-Axis PS: |f(x)| = |f(|x|)) in this case, it may not true in all cases

 f(x) is the output or the y-value of the function. |f(x)| therefore implies the modulus of the function itself Example if f(x) =4. |f(x)| = 4 and if  f(x) = -4, then |f(x)| =4 Hence , |f(x)| must lie above the X-axis. All the negative part of f(x) is reflected above the X-Axis to get |f(x)| f(x) = sinx |f(x)| = |sinx|

f(x) = |x| is a machine which accepts all kinds of numbers - positive, negative and 0 The positive numbers or 0, when enter the machine come out of it, without a change But when the negative numbers enter the machine, they are multiplied with -1 so as to make them positive. so, guess what happens when the input is "x"? What sign is x? What happens if x is positive? and What happens if x is negative?

The input of f(x) is x Whereas the input of f(|x|) is |x|  x=|x| when x is positive. Example f(x) = sinx f(|x|) = sin(|x|) Therefore graph of f(x) is same as that of f(|x|) for positive values of x only i.e on the right side of X-Axis x =-|x| for negative values of x. That implies if we input negative numbers in f(|x|), we end up giving -x as input and since x is negative in this case, -x is a positive number with the same value. Example when x=2 f(2) = f(|2|)  But if x=-2 f(|-2|) =f(2) Hence f(2)= f(|2|) = f(|-2|) When x=-3 f(|-3|) = f(3)  For x=-4 f(|-4|)=f(4) Whatever number, we may use, it turns positive before entering f(|x|) and hence is determined by f(x) where x is positive number Therefore f(|x|) is determined by f(x) with only positive inputs. For f(|x|) f(x)= f(-x) Hence whatever shape is f(|x|) is on right side of X-Axis as sketched previously, is on left side as well. f(|x|) is an even function.

 While the saints and all our ancient holy scriptures urge us to look within. Be it to find God or to cleanse your mind. All the faults lie within If the table shakes on which the cup of tea lies, it spills tea because it contains tea not because someone shook the table. That is how the perspectives are shaped when we follow saints. This is an honest truth that not everyone follows saints in the modern day and when democracy prevails. Some would like to think tea spilled because someone shook the table which is also correct in another frame. Whatever lies within the cup is none of our business, just don't spill it! What if democracy has reached a point when the "without" perspective of spilling the tea is widely prevalent? The essence of democracy is lost and it becomes dadagiri because we start demanding freedom that compromises the other's freedom. While democracy promotes the without perspective, the within perspective must be preserved to keep the democracy going....

Did I hear T is for te****s? "Every right implies a responsibility." I read the above quote from someone which is quite relevant in today's time when everyone is talking about rights and forgets that duties entail every right that we have.  After a year of requests, water wastage hasn't yet stopped. The arguments for wasting water They have the right to turn the motor on as they pay. But they want to justify not turning the same button off and "forgetting" despite having an alarm that rings when the tank fills up. Forget to turn the button off, Is that even an option? Therefore they are not at fault.  Yes, tomorrow you may say I press the accelerator of my car but I forget to take my foot off it.  C'mon, do it!

  We were 20 minutes to the airport in an Ola,   When Subhash’s Shabnam broke. Shubhash was our Ola driver for our journey to IGIA and his cab was lovingly called Shabnam.   Subhash seemed a cuckoo while loading our luggage in the cab. But we had no choice other than to board the cab. We crossed just a few steps from our home when it started raining cats and dogs from nowhere stopped at another mile and re-started in yet another mile with no sign of stopping this time. God was certainly amused. The water started pouring inside the cab with the rain. But it seemed harmless and Shubhash was behaving like a fool with his smiles trying to defend. We ignored it until he told us with the same sheepish smile that he was moving the fastest possible when the speed of his Shabnam was equal to that of a rickshaw. We suddenly got alerted. We yet have to cover a significant distance to reach our destination. Where will we be abandoned by Shubhash and his Shabnam? Will we get another c...

“Here is your room key Ma’am!” I asked “What is the room view" to which he told me there was a school outside. I again asked   “Any better view?” He said “The other rooms open to the road and hence city traffic, so this one is better”. I thought to myself city traffic? what are you talking about? I live in Delhi, it can’t be worse. “Alright,” I said to him and agreed to get the room No. 2## ### The room was nice, clean, and dimly lit (so if there are bugs they go unnoticed 🤭🤭) The next morning was a Sunday; we woke to delightful music being played in the school probably some hymns by a group of women. I was enjoying the music and then a guy started singing my then favorite song “Let Her Go” by Passenger when I lived in Mysore. The feeling of nostalgia hit me hard and I just couldn’t stop but open the window to look who was singing it. I could only see some people coming out of the school building. Nevertheless, that became my favorite memory of the Mysore trip which was compleme...