Which Best Describes the Domain of a Function

Which Best Describes the Domain of a Function.

A
function
is expressed as


y=f(x)
,

where
x
is the independent variable and
y
is the dependent variable.

First, we learn
what is the Domain
before learning

How to Notice the Domain of a Office Algebraically

πŸ‘‰
What is the Domain of a Function?

Let
f(x)
be a existent-valued role. Then the

domain of a part

is the set of all possible values of
x
for which
f(x)
is defined.

How to find the domain of a function algebraically

The domain of a function
f(x)
is expressed as
D(f).

We suggest y’all to read
how to discover zeros of a office
and
zeros of quadratic function
first.

πŸ‘‰
Rules to remember when finding the Domain of a Function

We should always remember the following rules when finding the domain of a function:

  1. If the function is a polynomial office then x tin be positive, zero or negative, i.east.,
    ten>0,
    x=0
    or
    ten<0,
  2. If
    f(x)=\frac{chiliad(10)}{h(ten)}, then always
    h(x)\neq0,
  3. If
    f(10)=\sqrt{k(x)}, then always
    g(x)\geq0,
  4. If
    f(10)=\frac{chiliad(x)}{\sqrt{h(x)}}, and then always
    h(ten)>0,
  5. If
    f(x)=\frac{\sqrt{g(x)}}{h(10)}, then always
    1000(10)\geq 0
    and
    h(x)\neq 0,
  6. If
    f(x)=\sqrt{\frac{chiliad(x)}{h(ten)}}, then always
    g(x)\geq0
    and
    h(ten)>0,
  7. If
    f(x)=\ln \left ( 1000(x) \right ), and then ever
    g(x)>0.

The vii rules mentioned above will make our piece of work like shooting fish in a barrel when we find the domain of a function.

There are 2 other rules. We will learn them at the time of discussion.

Table of Contents – What you lot will learn

πŸ‘‰
How to Find the Domain of a Function Algebraically

There are dissimilar ways to

find the domain of a function
.

Here we will discuss 9 best means for different functions.

#1. Find the Domain of a Polynomial Function

There are different types of
Polynomial Function
based on degree.

Some of them are

Degree of polynomial Office Polynomial Function Proper noun of Polynomial Office Domain
ane 3x+5 Linear Function \mathbb{R}=(-\infty,\infty)
ii xii+1 Quadratic Office \mathbb{R}=(-\infty,\infty)
three 2xiii+xii+3 Cubic Function \mathbb{R}=(-\infty,\infty)
four 7xiv+2 Quartic Role \mathbb{R}=(-\infty,\infty)
5 6xfive+2xii-three Quintic Role \mathbb{R}=(-\infty,\infty)
Domain of Polynomial Function

See that all the polynomial functions are defined for all
x\epsilon\mathbb{R}.

\therefore
the domain of whatsoever polynomial office is
\mathbb{R}=(-\infty,\infty).

#ii. Detect the Domain of a Rational Function

A
Rational Function
is a fraction of functions denoted by

f(10)=\frac{yard(ten)}{h(ten)}, h(x)\neq 0

Rational function is also chosen Quotient Function.

Example:

Permit
f(10)=\frac{x+two}{ten^{2}+3x+ii}. Observe the domain of
f(x).

Solution:

Run across that 10+2 is defined for all
x\epsilon \mathbb{R}.

So nosotros practise not have to worry for this part.

From Rule two we know that the function
f(x)=\frac{g(ten)}{h(10)}
is divers when
h(x)\neq0.

In this trouble, we have to find at what points
10^{2}+3x+2\neq 0.

Now
ten^{ii}+3x+2\neq 0

i.e.,
(10+2)(x+one)\neq 0

i.e., either
(x+2)\neq 0
or
(x+1)\neq 0

i.due east., either
ten\neq -ii
or
x\neq -one

Therefore
f(x)=\frac{ten+2}{x^{2}+3x+2}
exists for all
10\epsilon\mathbb{R}
except
x\neq -2

and
x\neq -one

\therefore
domain of f(ten) = {x\epsilon \mathbb{R}:ten\neq -2,-1}.

How to Find the Domain of a Function Algebraically - Best 9 Ways

On the Real centrality, the
light-green lines
are the domain of f(x).

We can write this equally,

Domain of f(x)=(-\infty,-two)\cup(-2,-ane)\cup(-ane,\infty).

Case:

How practise you find the domain of the rational function given beneath

f(ten)=\frac{x}{x^{two}+two}

Solution:

For f(ten) to be defined,

x^{ii}+2\neq 0

or,
ten^{2}\neq -two

or,
ten\neq \pm \sqrt{-two}

or,
x\neq \pm \sqrt{2}i \epsilon \mathbb{C}, an imaginary number (i.eastward., not a real number).

This implies that f(10) exists for all
x\epsilon \mathbb{R}

How to Find the Domain of a Function Algebraically - Best 9 Ways

\therefore
the domain of the function
f(x)=\frac{x}{x^{2}+ii}
is

D(f)=\mathbb{R}=(-\infty,\infty).

#3. Finding Domain of a Function with a Square Root

Case:

Observe the domain

f(x)=\sqrt{x+ii}

Solution:

From Rule 3 we know that a function of the form
f(x)=\sqrt{g(10)}
is defined when
one thousand(ten)\geq 0

i.e.,

f(x)=\sqrt{10+2}
is defined when

10+ii\geq 0

or,
x\geq -2

Putting this result on existent line nosotros get

How to Find the Domain of a Function Algebraically - Best 9 Ways

\therefore
domain of
f(x)=\sqrt{x+ii}
is

D(f)={x\epsilon \mathbb{R}: ten\geq -2}=[-two,\infty ).

Example:

Detect the domain of the function

f(x)=\sqrt{ten^{2}+3x+2}

Solution:

For
f(x)
to be defined,

10^{2}+3x+two\geq 0

i.e.,
(x+ii)(x+i)\geq 0

i.e., either
x+2\leq 0
or
10+i\geq 0

i.e., either
x\leq -two
or
x\geq -1

Why nosotros write
x\leq -2
and
x\geq -i?

See the table given below to understand this

Value of ten Sign of
(x+2)
Sign of
(10+i)
Sign of (x+2)(x+1) (x+ii)(x+i)\geq 0
satisfied or not
x=-three

i.e.,
x\epsilon (-\infty,-2)
-ve -ve >0
i.e., +ve
βœ…
ten=-2 0 -ve =0 βœ…
x=-i.five

i.e.,
x\epsilon (-ii,-1)
+ve -ve <0
i.e., -ve
❌
x=-1 +ve 0 =0 βœ…
x=0

i.e.,
x\epsilon (-one,\infty)
+ve +ve >0
i.e., +ve
βœ…

From the table we can see that the relation
(x+2)(10+i)\geq 0
is satisfied when

x\epsilon (-\infty,-2),
10=-2, and
10=-ane,
x\epsilon (-1,\infty)

i.e.,
10\epsilon (-\infty,-2]
and
10\epsilon [-1,\infty)

i.e.,
x\epsilon (-\infty,-two] \cup [-1,\infty)

How to Find the Domain of a Function Algebraically - Best 9 Ways

\therefore
domain of
f(ten)=\sqrt{10^{2}+3x+2}
is

D(f)
=

\: x \: \epsilon \: (-\infty,-ii] \cup [-one,\infty)

\: \: \: \: \: \: \: \: \: \:
= {ten \: \epsilon \: \mathbb{R}:x\leq -2,ten\geq -1}

#4. Finding Domain of a Part with a Square root in the denominator

From Rule 4 we know that a function of the form
f(x)=\frac{g(x)}{\sqrt{h(x)}}
is defined when
h(x)>0.

Example:

How to discover the domain of the function given below

f(x)=\frac{one}{\sqrt{i-x}}

Solution:

For
f(ten)
to exist defined,

1-x>0

i.due east.,
ane>10

i.due east.,
x<1

How to Find the Domain of a Function Algebraically - Best 9 Ways

\therefore
domain of
f(ten)={x\epsilon \mathbb{R}:x<ane} =
(-\infty,1)

Example:

Find the domain of

f(10)=\frac{ten^{2}+2x+three}{\sqrt{ten+1}}

Solution:

For
f(x)
to be defined,

x+one>0

i.e.,
10>-1

How to Find the Domain of a Function Algebraically - Best 9 Ways

\therefore
domain of
f(x)=\frac{x^{two}+2x+3}{\sqrt{ten+1}}
is {10\epsilon \mathbb{R}:10>-1} =
(-1,\infty)

#five. Finding Domain of a Part with a Square root in the numerator

From Rule 5 we know that a function of the form
f(10)=\frac{\sqrt{g(x)}}{h(x)}
is divers when
g(x)\geq 0
and
h(x)\neq 0.

Instance:

Find the domain of

f(x)=\frac{\sqrt{x+1}}{x^{2}-4}

Solution:

The function
f(x)=\frac{\sqrt{ten+1}}{10^{2}-4}
is defined when

  • x+1\geq 0
  • x^{2}-4\neq 0

Now

x+1\geq 0

i.east.,
10\geq -ane

and

10^{ii}-4\neq 0

i.eastward.,
(x+2)(ten-two)\neq 0

i.e.,
(x+two)\neq 0
and
(x-ii)\neq 0

i.e.,
x\neq -2
and
x\neq 2

How to Find the Domain of a Function Algebraically - Best 9 Ways

\therefore
domain of
f(x)=\frac{\sqrt{x+1}}{10^{ii}-four}
is

{10\epsilon \mathbb{R}:10\geq -one,x\neq 2,} (We doesn’t include
x\neq -2
because
x\geq -1)

We can also express the domain of the office in interval annotation.

Domain of
f(10)=\frac{\sqrt{x+i}}{x^{2}-4}
in interval annotation is
[-1,ii)\cup (2,\infty).

#half dozen. Finding Domain of a Function with a Square root in the numerator and denominator

From Rule half dozen we know that a function of the form
f(x)=\sqrt{\frac{g(x)}{h(ten)}}
is divers when
thou(ten)\geq0
and
h(ten)>0.

Example:

Find the domain of

f(ten)=\sqrt{\frac{x-2}{3-x}}

Solution:

For
f(10)
to be defined,

3-x\neq 0

i.e.,
iii\neq x

i.e.,
x\neq 3

At present we have to notice the set of values of x so that

\frac{x-2}{3-x}\geq 0

Hither we tin can non directly say
x-2>0
because we do non know the sign of
3-x.

To overcome this problem nosotros will brand the denominator +ve by multiplying the numerator and denominator by (iii-x)

\frac{x-2}{3-10}\times \frac{{\colour{Magenta} 3-x}}{{\color{Magenta} three-10}}\geq 0

i.e.,
\frac{(x-2)(iii-ten)}{(ten-3)^{two}}\geq 0

i.eastward.,
(x-2)(3-x)\geq 0

Next we accept to detect the values of x and so that
(x-2)(3-10)\geq 0

At present run across the tabular array given beneath:

Value of x Sign of (ten-2) Sign of (3-x) Sign of (x-2)(3-10) (x-ii)(3-x)\geq 0
satisfied or not
x=0

i.e.,
10\epsilon (-\infty,ii)
-ve +ve <0
i.e., -ve
❌
x=2 0 +ve 0 βœ…
ten=2.5

i.east.,
ten\epsilon (2,three)
+ve +ve >0
i.eastward., +ve
βœ…
x=iv

i.east.,
ten\epsilon (3,\infty)
+ve -ve <0
i.e., -ve
❌

Now putting the signs on real axis for each interval and value of x, nosotros become

How to Find the Domain of a Function Algebraically - Best 9 Ways

\therefore
the domain of the function
f(x)=\sqrt{\frac{ten-2}{iii-x}}
is
D(f)
=
[2,3)

#7. Observe Domain Of A Logarithmic Function

From Rule vii we know that a
Logarithmic Part
of the course
f(ten)=\ln \left ( thou(ten) \right )
is defined when
g(x)>0.

Instance:

Find the domain of

f(x)= \ln (x-2)

Solution:

The office
f(x)= \ln (x-ii)
is defined when

x-two>0

i.due east.,
ten>2

Therefore
f(x)= \ln (10-2)
is defined for all
x>2.

How to Find the Domain of a Function Algebraically - Best 9 Ways

\therefore
domain of
f(10)= \ln (x-2)
is

D(f)
= {ten\epsilon \mathbb{R}:x>2} =
(ii,\infty)

Example:

Detect the domain of

f(10)= \ln (x^{2}-3x+2)

Solution:

The function
f(x)= \ln (x^{2}-3x+ii)
is defined when

x^{2}-3x+2>0

i.e.,
(x-1)(x-2)>0

Value of 10 Sign of (x-1) Sign of (x-2) Sign of (ten-i)(10-2) (ten-i)(ten-two)>0 satisfied or non
x=.5<1
i.eastward.,
x\epsilon (-\infty,i)
-ve -ve +ve
i.e., >0
βœ…
x=i 0 -ve =0 ❌
x=1.five

i.e.,
10\epsilon (1,2)
+ve -ve -ve
i.e., <0
❌
ten=ii +ve 0 =0 ❌
x=3>2

i.e.,
x\epsilon (two,\infty)
+ve +ve +ve
i.e., >0
βœ…

i.e.,
10<1
and
10>2

How to Find the Domain of a Function Algebraically - Best 9 Ways

\therefore
domain of the part
f(x)= \ln (x^{2}-3x+2)
is

D(f)
= {ten\epsilon \mathbb{R}:x<1,x>2} =
(-\infty,1)\cup (2,\infty)

#viii. Find the Domain of a Part using a Relation

Rules:

  • Earlier finding the domain of a role using a relation first nosotros have to bank check that the given relation is a function or not,
  • A Relation is the prepare of ordered pairs i.e., the set of (x,y) and the domain of the relation is the set of all ten-coordinates i.e., the ready {ten}.

Case:

Find the domain of the relation

{(two,5), (3,6), (four,17), (11,8)}

Solution:

Showtime we check the relation {(ii,5), (iii,6), (4,17), (eleven,viii)} is a office or not.

The diagram of the relation is

Domain of a function using relation

See that each element of the ready {2, iii, 4, 11} is related to a unique element of the set {five, 6, 8, 17}.

Therefore the relation is a part.

In the relation

{(2,5), (iii,six), (4,17), (11,8)}

the set of x-coordinates is {2, 3, four, xi} and the prepare of y-coordinates is {v, vi, 17, 8}.

\therefore
the domain of the relation is {2, three, 4, xi}

Read more:
Which relation is a Function?

Example:

Discover the domain of the relation

{(2, iii), (5, 8), (6, 7), (six, fifteen), (11,17)}

Solution:

The diagram of the given relation is

Find the Domain of a function using relation

See that the element half-dozen is related to two different elements 7 and 15

i.due east., 6 is not related to a unique element.

Therefore the given relation is non a function.

Additional reading:
Which is relations are non a Function?

#nine. Find Domain of a Function on a Graph

Finding the domain of a function using a graph is the easiest way to find the domain.

Rule:

The domain of a function on a graph is the ready of all possible values of x on the ten-axis.

For domain, we have to find where the 10 value starts and where the 10 value ends i.due east., the function of x-axis where f(x) is divers.

See the example given below to understand this concept

Case:

Find the domain of the part from graph

x^{ii}+y^{2}=4.

Solution:

Step i: Draw the graph

How to Find the Domain of a Function Algebraically - Best 9 Ways

Step 2: Detect the possible values of ten where f(x) is defined

How to Find the Domain of a Function Algebraically - Best 9 Ways

Here the x values showtime from -ii and ends in 2.

Step 3: The possible values of ten is the domain of the function.

How to Find the Domain of a Function Algebraically - Best 9 Ways

\therefore
the domain of the circle is {x\epsilon \mathbb{R}:-ii\leq x\leq 2} =
[-ii,2]

Example:

Notice the domain of the part

y^{two}=2(x-two)

Solution:

Step ane: The graph of the given parabola is

How to Find the Domain of a Function Algebraically - Best 9 Ways

Pace 2:

See that the 10 value starts from 2 and extends to infinity (i.due east., it will never stop).

How to Find the Domain of a Function Algebraically - Best 9 Ways

Stride three:

\therefore
the domain of the parabola is {x\epsilon \mathbb{R}:2\leq ten\leq \infty} =
[ii,\infty)

How to Find the Domain of a Function Algebraically - Best 9 Ways

Example:

Find the domain of the direct line
y=ten
from the graph

How to Find the Domain of a Function Algebraically - Best 9 Ways

Solution:

From the graph of
y=x
we tin can come across that the ten value starts from
-\infty
and extends to
+\infty.

How to Find the Domain of a Function Algebraically - Best 9 Ways

Therefore the domain of the straight line is
(-\infty,\infty).

Example:

Find the domain of
x^{2}=2y
from the graph given below.

How to Find the Domain of a Function Algebraically - Best 9 Ways

Solution: The x values of
x^{2}=2y
on the graph are shown by the green line.

How to Find the Domain of a Function Algebraically - Best 9 Ways

See that the
x
value starts from
-\infty
and extends to
+\infty.

Therefore domain of
10^{2}=2y
is
(-\infty,\infty).

Final words

We just learned 9 different ways to
Notice the Domain of a Function Algebraically.

Now it’southward your plough to practice them over again and again and master them.

If you have any doubts or suggestions, please tell us in the comment section. We love to hear from you.

Additionally you lot tin can read:

  • What is a Function?
  • What are the 48 Types of Functions?
  • How to Find the Zeros of a Function?
  • What is the Limit of a Office?
  • 13 All-time means to Find the Limit of a Function?
  • How to Find the Limit using Squeeze Theorem?

Which Best Describes the Domain of a Function

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