How To Find Domain & Range -Part-II

In general, we determine the domain of each function by looking for the values of independent variable (usually x) which we are allowed to use. We avoid 0 in the denominator, or negative values under square root sign. Because if 0 will come in the denominator than function will not be defined so we exclude these numbers from the real numbers. Similarly negative sign in the square root is not defined for real valued.

When finding the domain, remember:

• The denominator of a fraction can not be zero.
• The number under a square root sign must be positive.

Range of a function is the set of all possible resulting values of the dependent variable (usually y), after we have substituted the domain in the given function.

• The range of a function is the possible minimum y values to maximum y values.
• Substitute different x values into the expression for y to see what is happening. (Ask yourself: Is y always positive? Always negative? or may be equal to certain values ?)
• Make sure you are looking for minimum and maximum values of y
• Draw the graph of the function, from the graph you can see clearly see the range.

Let us consider one example to find domain and range?

y = f(x) = √(16 -x²)

For domain,

⇒ 16 -x²≥0

⇒ x² -16≤ 0

⇒ (x -4)(x+4) ≤ 0

⇒ -4≤ x ≤ 4

⇒x ∈ [ -4 ,4]

Hence, Domain(f)=[-4,4]

For Range, f(x) = y =√( 16 -x²)

⇒ y²=16 -x²

⇒ x²=16 -y²

⇒ x =√( 16 -y²)

Clearly, x will take real values, if

⇒ 16 -y² ≥ 0

⇒ y² -16 ≤ 0

⇒ (y -4)(y+4) ≤ 0

⇒ -4 ≤ y ≤ 4

⇒ y ∈ [-4,4 ]

Also, y=√( 16 -x²) ≥ 0 for all x ∈ [ -4 ,4]

Therefore,y ∈ [ 0,4] for all x ∈ [ -4 ,4]

Hence, Range(f) = [0 ,4]

Let us consider another example, y =1/√(x -5) Find domain and range?

we have, y =1/√(x -5)

Clearly, given function takes real values, if

⇒ x-5 > 0

⇒ x >5

⇒ x ∈ ( 5,∞)

⇒ domain(f) = ( 5,∞)

For x>5 ,we have

x -5> 0 ⇒ √(x -5)> 0

⇒ 1/√(x -5) > 0

⇒ y>0

Hence, function takes all real values greater than zero.

Range(f) = ( 0,∞)