FINDING DOMAIN AND RANGE OF MODULUS FUNCTION

Question 1 :

Find the domain and range of the following functions.

f(x)  =  |x - 3|

Solution :

Domain :

A set of all defined values of x is known as domain.

Range :

The out comes or values that we get for y is known as range.

Domain for given function f(x)  =  |x - 3|

For any real values of x, f(x) will give defined values. Hence the domain is R.

Since we have absolute sign, we must get only positive values by applying any positive and negative values for x in the given function.

So, the range is [0, ∞).

Question 2 :

Find the domain and range of the following functions.

f(x)  =  1 - |x - 2|

Solution :

For any values values of x, the function will give defined values. It will never become undefined.

So, domain is all real values that is R.

The range of |x - 2| lies between 0 to ∞. But we need find the range of 1 - |x - 2|

0 ≤ |x - 2| ≤ ∞

Multiplying by negative through out the inequality, we get

- ∞ ≤ -|x - 2| ≤ 0

Add 1-, through out the inequality, we get

1 - ∞ ≤ -|x - 2| ≤ 1 - 0

- ∞ ≤ -|x - 2| ≤ 1 

So, the range is (- ∞, 1].

Question 3 :

Find the domain and range of the following functions.

f(x)  =  |x - 4|/(x - 4)

Solution :

x - 4  =  0

x  =  4

Domain is R - {4}

In order to find range, we may split the given function as two parts.

So, range is [-1, 1].

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