Domain of a Function: Multiple domains

Find the domain of the functionFinding the Domain of a Square Root FunctionFind the domain of an expression with fractions
Question 1

Find the area of domain (no need to solve)

\( 9(4x-\frac{5}{x})=20(3x-\frac{6}{x+1}) \)

Question 2

Find the area of domain (no need to solve)

\( \frac{7}{x+5}=\frac{6}{13x} \)

Question 3

Find the area of domain (no need to solve)

\( \frac{14}{x}-6x=\frac{2}{x-5} \)

Question 4

Find the area of domain (no need to solve)

\( (\frac{4}{x-2})\times(\frac{7x}{x-6})=2 \)

Question 5

Find the domain

(no need to resolve)

\( \frac{5x}{2(x-7)}=\frac{10}{8x} \)

Examples with solutions for Domain of a Function: Multiple domains

Exercise #1

Find the area of domain (no need to solve)

9(4x−5x)=20(3x−6x+1) 9(4x-\frac{5}{x})=20(3x-\frac{6}{x+1})

Video Solution

Step-by-Step Solution

The domain of the equation is the set of domain values (of the variable in the equation) for which all algebraic expressions in the equation are well defined,

From this, of course - we exclude numbers for which arithmetic operations are not defined,

In the expression on the left side of the given equation:

9(4x−5x)=20(3x−6x+1) 9(4x-\frac{5}{x})=20(3x-\frac{6}{x+1})

there is multiplication between fractions whose denominators contain algebraic expressions that include the variable of the equation (which we are looking for when solving the equation),

Of course, these fractions are defined as long as the expressions in their denominators are not equal to zero (since division by zero is not possible),

Therefore, the domain of definition of the variable in the equation will be obtained from the requirement that these expressions (in the denominators of the fractions) do not equal zero, meaning:

For the fraction in parentheses in the expression on the left side we get:

x≠0 \boxed{ x\neq0}

For the fraction in parentheses in the expression on the right side we get:

x+1≠0 x+1\neq0 \\ Let's solve the second inequality above (in the same way as solving an equation):

x+1≠0x≠−1 x+1\neq0 \\ \boxed{x\neq-1}

Therefore, the correct answer is answer A.

Note:

It should be noted that the above inequality is a point inequality and not a trend inequality (meaning it negates equality: (≠) (\neq) and does not require a trend: (<,>,\leq,\geq) ) which is solved exactly like solving an equation, this is unlike solving a trend inequality where different solution rules apply depending on the type of expressions in the inequality, for example: solving a first-degree inequality with one variable (which has only first-degree and lower algebraic expressions), is solved almost identically to solving an equation, however any division or multiplication of both sides by a negative number requires reversing the trend.

Answer

x≠0,x≠−1 x≠0,x≠-1

Exercise #2

Find the area of domain (no need to solve)

7x+5=613x \frac{7}{x+5}=\frac{6}{13x}

Video Solution

Step-by-Step Solution

The domain of the equation is the set of domain values (of the variable in the equation) for which all algebraic expressions in the equation are well defined,

From this, of course - we exclude numbers for which arithmetic operations are not defined,

In the expression on the left side of the given equation:

7x+5=613x \frac{7}{x+5}=\frac{6}{13x}

there is multiplication between fractions whose denominators contain algebraic expressions that include the variable of the equation (which we are looking for when solving the equation),

Of course, these fractions are defined as long as the expression in their denominators is not equal to zero (since division by zero is not possible),

Therefore, the domain of definition of the variable in the equation will be obtained from the requirement that these expressions (in the denominators of the fractions) do not equal zero, meaning:

For the fraction in the expression on the left side we get:

x+5≠0 x+5\neq0 \\ For the fraction in the expression on the right side we get:

13x≠0 13x\neq0

We will solve these inequalities (in the same way as solving an equation):

x+5≠0x≠−5 x+5\neq0 \\ \boxed{x\neq-5}

13x≠0/:13x≠0 13x\neq0 \hspace{8pt}\text{/:13} \\ \boxed{x\neq0}

Therefore, the correct answer is answer A.

Note:

It should be noted that the above inequality is a point inequality and not a directional inequality (meaning it negates equality: (≠) (\neq) and does not require direction: (<,>,\leq,\geq) ) which is solved exactly like solving an equation, this is unlike solving a directional inequality where different solution rules apply depending on the type of expressions in the inequality, for example: solving a first-degree inequality with one variable (which only has first-degree algebraic expressions and below), is solved almost identically to solving an equation, however, any division or multiplication of both sides by a negative number requires reversing the direction.

Answer

x≠0,x≠−5 x≠0,x≠-5

Exercise #3

Find the area of domain (no need to solve)

14x−6x=2x−5 \frac{14}{x}-6x=\frac{2}{x-5}

Video Solution

Answer

x≠0,x≠5 x≠0,x≠5

Exercise #4

Find the area of domain (no need to solve)

(4x−2)×(7xx−6)=2 (\frac{4}{x-2})\times(\frac{7x}{x-6})=2

Video Solution

Answer

x≠2,x≠6 x≠2,x≠6

Exercise #5

Find the domain

(no need to resolve)

5x2(x−7)=108x \frac{5x}{2(x-7)}=\frac{10}{8x}

Video Solution

Answer

x≠0,x≠7 x≠0,x≠7

Question 1

Given the following function:

\( \frac{\sqrt{3x^2+3}}{9} \)

What is the domain of the function?

Question 2

Given the following function:

\( \frac{10x+2}{\sqrt{x^2-4}} \)

What is the domain of the function?

Question 3

Look at the following function:

\( \frac{5x+4}{\sqrt{x^2-9}} \)

What is the domain of the function?

Question 4

Look at the following function:

\( \frac{8x}{\sqrt{2x^2-2}} \)

What is the domain of the function?

Question 5

Look at the following function:

\( \frac{\sqrt{2.5x^2-5}}{5} \)

What is the domain of the function?

Exercise #6

Given the following function:

3x2+39 \frac{\sqrt{3x^2+3}}{9}

What is the domain of the function?

Video Solution

Answer

The entire domain

Exercise #7

Given the following function:

10x+2x2−4 \frac{10x+2}{\sqrt{x^2-4}}

What is the domain of the function?

Video Solution

Answer

x > 2,x < -2

Exercise #8

Look at the following function:

5x+4x2−9 \frac{5x+4}{\sqrt{x^2-9}}

What is the domain of the function?

Video Solution

Answer

x > 3,x < -3

Exercise #9

Look at the following function:

8x2x2−2 \frac{8x}{\sqrt{2x^2-2}}

What is the domain of the function?

Video Solution

Answer

x > 1,x < -1

Exercise #10

Look at the following function:

2.5x2−55 \frac{\sqrt{2.5x^2-5}}{5}

What is the domain of the function?

Video Solution

Answer

x≥2,x≤−2 x\ge\sqrt{2},x\le-\sqrt{2}

Question 1

Look at the following function:

\( \frac{\sqrt{-3x^2+12}}{9} \)

What is the domain of the function?

Question 2

Look at the following function:

\( \frac{\sqrt{3x^2+7}}{12} \)

What is the domain of the function?

Question 3

Look at the following function:

\( \frac{\sqrt{4x^2-4}}{10} \)

What is the domain of the function?

Question 4

Look at the following function:

\( \frac{\sqrt{4x^2-8}}{5} \)

What is the domain of the function?

Question 5

Look at the following function:

\( \frac{\sqrt{5x^2+2}}{10} \)

What is the domain of the function?

Exercise #11

Look at the following function:

−3x2+129 \frac{\sqrt{-3x^2+12}}{9}

What is the domain of the function?

Video Solution

Answer

−2≤x≤2 -2 \le x \le 2

Exercise #12

Look at the following function:

3x2+712 \frac{\sqrt{3x^2+7}}{12}

What is the domain of the function?

Video Solution

Answer

All real numbers

Exercise #13

Look at the following function:

4x2−410 \frac{\sqrt{4x^2-4}}{10}

What is the domain of the function?

Video Solution

Answer

x≥1,x≤−1 x\ge1,x\le-1

Exercise #14

Look at the following function:

4x2−85 \frac{\sqrt{4x^2-8}}{5}

What is the domain of the function?

Video Solution

Answer

x≥2,x≤−2 x\ge\sqrt{2},x\le-\sqrt{2}

Exercise #15

Look at the following function:

5x2+210 \frac{\sqrt{5x^2+2}}{10}

What is the domain of the function?

Video Solution

Answer

All real numbers

Question 1

Look at the following function:

\( \frac{\sqrt{x^2+2}}{3} \)

What is the domain of the function?

Question 2

Look at the following function:

\( \frac{x+2}{\sqrt{3x^2-9}} \)

What is the domain of the function?

Question 3

Find the area of domain (no need to solve)

\( \frac{x}{5x-6}=\frac{2}{x-1} \)

Question 4

Does the following equation have a true or false value?

\( \frac{x^2-81}{(x-9)(x+9)}=1 \)

Question 5

\( \frac{\sqrt{15}+34:z}{4y-12+8:2}=5 \)

What is the field of application of the equation?

Exercise #16

Look at the following function:

x2+23 \frac{\sqrt{x^2+2}}{3}

What is the domain of the function?

Video Solution

Answer

All real numbers

Exercise #17

Look at the following function:

x+23x2−9 \frac{x+2}{\sqrt{3x^2-9}}

What is the domain of the function?

Video Solution

Answer

x>\sqrt{3},x<-\sqrt{3}

Exercise #18

Find the area of domain (no need to solve)

x5x−6=2x−1 \frac{x}{5x-6}=\frac{2}{x-1}

Video Solution

Answer

x≠1,x≠115 x≠1,x≠1\frac{1}{5}

Exercise #19

Does the following equation have a true or false value?

x2−81(x−9)(x+9)=1 \frac{x^2-81}{(x-9)(x+9)}=1

Video Solution

Answer

True only when x≠±9 x\ne\pm9 .

Exercise #20

15+34:z4y−12+8:2=5 \frac{\sqrt{15}+34:z}{4y-12+8:2}=5

What is the field of application of the equation?

Video Solution

Answer

y≠⁡2 y\operatorname{\ne}2