Domain of the function — lesson. Mathematics State Board, Class 10.

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In this topic, we are going to consider the function as a real-valued function.

The domain of a function is the set of all possible input values that produce some output value range.

While finding the domain, do remember the following things:

The denominator (bottom) of a fraction cannot be zero.
The number under a square root sign must be non-negative.

Example:

Consider the following real value function and find its domain.

$f(x) =$

$1-x$

This function takes all possible real values of

$x$ . Thus, the domain of the function

$f$ is the set of all real numbers

$\mathbb{R}$ .

$f(x) =$

$\frac{1}{x}$ .

This function is in fraction form.

The denominator of a fraction cannot be zero. That is,

$x$ cannot be

$0$ .

The function takes all possible real values of

$x$ except

$0$ . Thus, the domain of the function

$f$ is the set of all real numbers

$\mathbb{R} - \{0\}$ .

$f(x) =$

$\frac{1}{x+10}$ .

This function is in fraction form.

The denominator of a fraction cannot be zero. That is,

$x+10$ cannot be

$0$ .

Suppose

$x+10 = 0$ .

Then

$x = -10$ .

Therefore,

$x$ cannot be

$-10$ .

The function takes all possible real values of

$x$ except

$-10$ . Thus, the domain of the function

$f$ is the set of all real numbers

$\mathbb{R} - \{-10\}$ .

$f(x) =$

$\frac{3x+2}{x^2+5x+6}$ .

This function is in fraction form.

So denominator of a fraction cannot be zero. That is,

$x^2+5x+6$ cannot be

$0$ .

Suppose

$x^2+5x+6 = 0$ .

Factorising the equation, we have

$x = -2, x = -3$ .

Therefore,

$x$ cannot be

$-2, -3$ .

The function takes all possible real values of

$x$ except

$-2$ and

$-3$ . Thus, the domain of the function

$f$ is the set of all real numbers

$\mathbb{R} - \{-2,-3\}$ .

$f(x) =$

$\sqrt{x^2-9}$ .

This function is in square root form.

So the number under square root should not be negative. That is,

$x^2-9$ cannot be negative.

This results in a negative only if the value of

$x$ is greater than

$3$ or less than

$-3$ .

The function takes all possible real values from

$-3$ to

$3$ , Thus, the domain of the function

$f$ is the set of all real numbers in the interval

$[-3,3]$ .

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3. Domain of the function

Theory: