**Domain and Range of a Function Worksheet : **

Worksheet given in this section will be much useful for the students who would like to practice problems on finding domain and range of a function.

Before look at the worksheet, if you would like to learn, how to find domain and range of a function,

**Problem 1 :**

Find the domain and range of the function :

y = 2x + 1

**Problem 2 :**

Find the domain and range of the function :

y = 1 / (x-2)

**Problem 3 :**

Find the domain and range of the function :

y = √(3x+1)

**Problem 1 :**

Find the domain and range of the function :

y = 2x + 1

**Solution :**

**Domain :**

To find the domain of a function, we have to find the values of x for which the given function is defined.

The above function is defined for all real values of x.

Therefore, the domain of the given function is all real values or R.

**Range : **

To find the range of a function, first we have to find the inverse of y.

To find inverse of y, follow the steps given below.

Step 1 :

y = 2x + 1 is defined by y in terms x.

The same function has to be redefined by x in terms of y.

Step 2:

y = 2x + 1

Subtract 1 from each side.

y - 1 = 2x

Divide each side by 2.

(y-1) / 2 = x

x = (y-1) / 2

Now the function has been defined by x in terms of y.

Step 3 :

In x = (y-1) / 2, we have to replace x by y^{-1} and y by x.

Then,

y^{-1} = (x-1) / 2

Step 4 :

Now, find the domain of y^{-1}.

The inverse function y^{-1} is defined for all real values of x.

So, the domain of y^{-1} is all real values or R.

And we already know the fact that

Range (y) = Domain (y^{-1})

Therefore, the range of the given function is all real values or R.

**Problem 2 :**

Find the domain and range of the function :

y = 1 / (x-2)

**Solution :**

**Domain :**

To find the domain of a function, we have to find the values of x for which the given function is defined.

In the above function, if we substitute 2 for x, then we get

y = 1/(2-2)

y = 1/0

y = Undefined

The denominator becomes zero and the function becomes undefined for x = 2.

So, the above function is defined for all real values of x except 2.

Therefore, the domain of the given function is

R - {2}

**Range : **

To find the range of a function, first we have to find the inverse of y.

To find inverse of y, follow the steps given below.

Step 1 :

y = 1 / (x-2) has been defined by y in terms x.

The same function has to be redefined by x in terms of y.

Step 2:

y = 1 / (x-2)

Multiply each side by (x-2).

(x-2)y = 1

xy - 2y = 1

Add 2y to each side.

xy = 2y + 1

Divide each side by y.

x = (2y+1) / y

Now the function has been defined by x in terms of y.

Step 3 :

In x = (2y+1) / y, we have to replace x by y^{-1} and y by x.

Then,

y^{-1} = (2x+1) / x

Step 4 :

Now, find the domain of y^{-1}.

In the inverse function y^{-1}, if we substitute 0 for x, the denominator will become zero.

That is, y^{-1} is undefined for x = 0.

Then, y^{-1} is defined for all real values of x except zero.

So, the domain of y^{-1} is

R - {0}

And we already know the fact that

Range (y) = Domain (y^{-1})

Therefore, the range of the given function is

R - {0}

**Problem 3 :**

Find the domain and range of the function :

y = √(3x+1)

**Solution :**

**Domain :**

To find the domain of a function, we have to find the values of x for which the given function becomes undefined.

The above function has square root.

If the function y = √(3x+1) is defined, then the value inside the square root has to be either zero or positive.

So, the function y = √(3x+1) is defined, when (3x + 1) ≥ 0

Then,

3x + 1 ≥ 0

Subtract 1 to each side.

3x ≥ -1

Divide each side by 3.

x ≥ -1/3

So, the above function is defined for x ≥ -1/3.

Therefore, the domain of the given function is

[-1/3, +∞)

**Range : **

In the given function y = √(3x+1), there is positive square root on the right side.

So, the value of y is either zero or positive.

Therefore, the range of the given function is

[0, +∞)

After having gone through the stuff given above, we hope that the students would have understood, how to find domain and range of a function.

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