Domain - Examples, Exercises and Solutions

Understanding Domain

Complete explanation with examples

The transposition of terms involves passing the terms of an equation from one member to another. In fact, it is a group of numbers that, according to mathematical rules, are allowed to be placed in place of the unknown (or variable) within an equation. The concept of transposing terms is especially concerning equations with fractions or square roots in order to find the domain of the equation.

In certain cases we must pay attention to the transposition of terms:

  1. In case of fraction the denominator cannot be equal to zero.
  2. In case of square root, the radicand cannot be negative.

This means that, in such cases, it is not enough to solve the equation, but we must check if the given solution is sustainable or makes sense in the real numbers.

How do you find the domain of an equation with one unknown?
1. In equations with fractions we will find the domain by equaling the denominator to zero.
The value or values that cause the denominator to equal zero are outside the domain of the equation.

Mathematical function F(X) = 1/X. Explanation of why X ≠ 0 due to division by zero being undefined. Fundamental algebra and function domain restriction concept.

  • In an equation with a root the values that cause the root to be negative are outside the domain of the equation.
Mathematical function F(X) = √X. Explanation that a square root cannot be negative, leading to the domain restriction X ≥ 0. Fundamental concept in algebra and function domains.

Below we can see an example of how to find the domain of an equation using transpose of terms:

Example :

1(X2)=1 \frac{1}{(X-2)}=1

This is an equation with fraction in which the unknown appears in the denominator. The denominator cannot be zero, so the expression is not well defined:

0=X2 0=X-2

Using transposition of terms we can clear the unknown and we obtain:

X=2 X=2

Therefore, the domain of the function is all real numbers except when X=2 X=2 .

Detailed explanation

Practice Domain

Test your knowledge with 7 quizzes

\( \frac{x+y:3}{2x+6}=4 \)

What is the field of application of the equation?

Examples with solutions for Domain

Step-by-step solutions included
Exercise #1

Select the field of application of the following fraction:

x16 \frac{x}{16}

Step-by-Step Solution

Let's examine the given expression:

x16 \frac{x}{16}

As we know, the only restriction that applies to a division operation is division by 0, since no number can be divided into 0 parts, therefore, division by 0 is undefined.

Therefore, when we talk about a fraction, where the dividend (the number being divided) is in the numerator, and the divisor (the number we divide by) is in the denominator, the restriction applies only to the denominator, which must be different from 0,

However in the given expression:

x16 \frac{x}{16}

the denominator is 16 and:

160 16\neq0

Therefore the fraction is well defined and thus the unknown, which is in the numerator, can take any value,

Meaning - the domain (definition range) of the given expression is:

all x

(This means that we can substitute any number for the unknown x and the expression will remain well defined),

Therefore the correct answer is answer B.

Answer:

All X All~X

Video Solution
Exercise #2

Select the domain of the following fraction:

8+x5 \frac{8+x}{5}

Step-by-Step Solution

The domain depends on the denominator and we can see that there is no variable in the denominator.

Therefore, the domain is all numbers.

Answer:

All numbers

Video Solution
Exercise #3

Select the the domain of the following fraction:

6x \frac{6}{x}

Step-by-Step Solution

The domain of a fraction depends on the denominator.

Since you cannot divide by zero, the denominator of a fraction cannot equal zero.

Therefore, for the fraction 6x \frac{6}{x} , the domain is "All numbers except 0," since the denominator cannot equal zero.

In other words, the domain is:

x0 x\ne0

Answer:

All numbers except 0

Video Solution
Exercise #4

2x3=4x 2x-3=\frac{4}{x}

What is the domain of the exercise?

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the fraction's denominator.

  • Step 2: Determine where this denominator equals zero.

  • Step 3: Exclude this value from the domain.

Now, let's work through each step:

Step 1: The given equation is 2x3=4x 2x - 3 = \frac{4}{x} . Notice that the fraction 4x\frac{4}{x} has a denominator of xx.

Step 2: Set the denominator equal to zero to determine where it is undefined.

xamp;=0 \begin{aligned} x &= 0 \end{aligned}

Step 3: Since the expression is undefined at x=0x = 0, we must exclude this value from the domain.

Therefore, the domain of the expression is all real numbers except 0, formally stated as x0 x \neq 0 .

The correct solution to the problem is: x ≠ 0.

Answer:

x≠0

Video Solution
Exercise #5

2x+6x=18 2x+\frac{6}{x}=18

What is the domain of the above equation?

Step-by-Step Solution

To solve this problem and find the domain for the expression 2x+6x2x + \frac{6}{x}, we apply the following steps:

  • Step 1: Identify when the fraction 6x\frac{6}{x} is undefined. This occurs when the denominator xx equals zero.
  • Step 2: To find the restriction, set the denominator equal to zero: x=0x = 0.
  • Step 3: Solve for xx to find the values excluded from the domain. Here, x0x \neq 0.

Since 6x\frac{6}{x} is undefined for x=0x = 0, the value x=0x = 0 must be excluded from the domain.
Hence, the domain of the equation is all real numbers except zero.

Therefore, the solution to the problem, indicating the domain of the expression, is x0 x \neq 0 .

Answer:

x≠0

Video Solution

More Domain Questions

Continue Your Math Journey

Topics Learned in Later Sections

Practice by Question Type