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Modulus Function

y = |x|
 
It is defined as
 
y x; if x >0
y = −x; if x <0
y = 0; if x = 0

Despite that in the above equations, we are finding a negative value of x if x < 0, its absolute value can never be negative. This can be seen through the following example:
 
We are finding out the value of y = |−5|.
 
Assuming that – 5 = x, so y = |x|
 
Now, since x <0, so y = −x = −(−5) = 5

Graphical representation of modulus function
 
This is the graph of y = |x|
 
Description: 10-2.tif

It is observed that for every value of x, be it +ve or –ve, the value of y cannot be negative.

 
Example-1
What is the value of x if |2x + 3| = 9?
Solution
In the questions involving modulus, first the value of expression under the modulus is taken as a positive and then as a negative.
Case 1 When (2x + 3) > 0, or, x > Description: 2469.png, then |2x + 3|
= 2x + 3
So, 2x + 3 = 9 or, 2x = 6.
So, x = 3
 
Case 2 When (2x + 3) < 0, or, x < Description: 2478.png, then |2x + 3|
= −(2x + 3)
So, −(2x + 3) = 9, or, −2x = 12
So, x = −6
 
 
Example-2
What is the value of ‘x’ if x2 + 5|x| + 6 = 0?
Solution
Taking x > 0, x2 + 5|x| + 6 = x2 + 5x + 6 = (x + 2) (x + 3) = 0
Or, x = −2 and x = −3
 
But as we have assumed that x>0, so x = −2 and –3 are not admissible.
 
Taking x < 0, x2 + 5|x| + 6 = x2 − 5x + 6 = (x − 2) (x − 3) = 0
Or, x = 2 and x = 3
 
But as we have assumed that x<0, so x = 2 and 3 are not admissible.
 
So, there is no real value of x which can satisfy this equation.
 
Alternatively, it can be seen that x2 and 5|x| and 6, are all positive values. So the sum of these three can never be equal to 0, so no real value of x is possible.
 
 
Example-3
If |x2 – 5x + 6| > x2 – 5x + 6, then find the values of x?
Solution
If |N| > N, then N < 0. (It can be understood by assuming the values)
 
So, x2 – 5x + 6 < 0
 
Or, (x – 2) (x – 3) < 0
 
So, 2 < x < 3
 

Greatest Integer Value Function
y = [x]
 
It is defined as the largest integral value of x which is less than or equal to x.
 
It is given y = [3.23] and we have to find the greatest integer value of y.
 
Taking the second part of the definition, i.e., the integer less than or equal to 3.23, we get a set of integers less than or equal to 3.23 ⇒ 3, 2, 1, 0, −1,…and so on.
 
The largest integer among all these integers = 3. So the greatest integer value of [3.23] = 3
 
Similarly, if we find the greatest integer value of y = [−2.76], then all the integers less than this value (–2.76) = {−3, −4, −5, −6,…}.
 
Now the greatest integer among all these integers given in the above set = −3
 
It can also be seen through the tabular presentation:
 

x

y

0–1 (excluding at x = 1)

0

1–2 (excluding at x = 2)

1

2–3 (excluding at x = 3)

2

 
And so on…
Example-4
What is the value of x in the following expression?
[x]2 ≤ 16?
Solution
4 ≤ [x] ≤ 4
 
−4 ≤ [x], or, −4 ≤ x
 
And [x] ≤ 4, or, x < 5.
 
So, the value of x is: −4 ≤ x < 5.
 

Logarithmic Function

y = logax
 
It is known that the value of x has to be positive here. However, y can have negative values.
 
Since with an increase in the value of xy always increases; so y = logax is an increasing function.
 
We will discuss more about this function in the logarithm chapter.

Exponential Function

y = ex

Exponential function is the inverse of a logarithmic function.

Again it can be observed that the value of y cannot be negative, whatever be the value of x in y = ex

Constant Function

f(x) = k, (where k is any constant) is known as a constant function.
 
Description: 10-3.tif





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