Domain — Introduction (What is a Domain)

Domain — Introduction (What is a Domain). Practice questions to deepen understanding of the domain of a function — introduction. Online math practice with full solutions and step-by-step explanations.

This practice covers the definition of a domain and the three restrictions: division by zero, root of a negative number, logarithm.

Domain introduction practice — what a domain is, the three restrictions: division by zero, root of a negative number, logarithm. Explanations for beginners.

Definition: what is a domain. Restriction #1: division by zero (≠ 0). Restriction #2: root of a non-negative.

10 questions

Question 1

10.00 pts

📚 What is a domain?

What is the definition of the domain of a function?

The set of all \(x\) values for which the function is defined and returns a real value

The set of all possible \(y\) values

All real numbers

Only the positive numbers

Explanation:

📚 Domain of definition — the definition

Definition:

Domain of definition of a function:

The set of all values of \(x\)
for which the function is defined
and returns a real value

In simple terms:

"Which values of x are allowed to substitute?"

• Allowed = in domain ✓
• Not allowed = not in domain ✗

Real-life example:

🎢 Roller coaster:

Minimum height requirement: 1.20 m

Domain: [1.20, ∞)

If your height < 1.20 → not defined!
(cannot board)

Mathematical notation:

Domain = \(D_f\) or Domain

Examples:
\(D_f = \mathbb{R}\) (all reals)
\(D_f = [0, \infty)\) (non-negative)
\(D_f = \mathbb{R} \setminus \{3\}\) (all except 3)

Question 2

10.00 pts

⚠️ The first restriction:

Why is division by zero not allowed?

Division by zero is not defined in mathematics — there is no unique answer

It gives infinity

It gives zero

It is always allowed

Explanation:

⚠️ Restriction #1: division by 0

The rule:

\(\frac{a}{0}\) = not defined! ⛔

The denominator must be \(\neq 0\)

Why?

Suppose \(\frac{6}{0} = x\)

then: \(6 = 0 \cdot x\)

but \(0 \cdot x = 0\) always!

No number satisfies the equation ❌

Real-life example:

🍕 Sharing a pizza:

You have 8 slices of pizza

Want to divide among 0 people?

This does not make sense! 🤷

There is no meaning to such a division

Examples:

\(f(x) = \frac{1}{x}\)
Domain: \(\mathbb{R} \setminus \{0\}\)

\(f(x) = \frac{1}{x-3}\)
Domain: \(\mathbb{R} \setminus \{3\}\)

\(f(x) = \frac{1}{x^2-4}\)
Domain: \(\mathbb{R} \setminus \{-2, 2\}\)

Question 3

10.00 pts

√ The second restriction:

What is the condition for \(\sqrt{x}\)?

\(x \geq 0\) — only non-negative (including 0)

\(x > 0\) — only strictly positive

\(x \neq 0\)

All numbers are allowed

Explanation:

√ Restriction #2: square root of a negative

The rule:

\(\sqrt{x}\) is defined only if:

\(x \geq 0\)

⚠️ Note: including 0!

Why?

Square root:
"Which number squared gives x?"

The square of a real number is always ≥ 0

So there is no real square root for a negative number! ❌

What about 0?

\(\sqrt{0} = 0\) ✓

It is fully defined!

because \(0^2 = 0\)

Examples:

\(f(x) = \sqrt{x}\)
Domain: \([0, \infty)\) ✓

\(f(x) = \sqrt{x-5}\)
Domain: \([5, \infty)\) ✓

\(f(x) = \sqrt{-x}\)
Domain: \((-\infty, 0]\) ✓

Real-life example:

❄️ Freezer:

Minimum temperature: -18°
Cannot freeze below that

Domain: \([-18, \infty)\)

Question 4

10.00 pts

📊 The third restriction:

What is the condition for \(\log(x)\)?

\(x > 0\) — only strictly positive (not including 0!)

\(x \geq 0\)

\(x \neq 0\)

All numbers are allowed

Explanation:

📊 Restriction #3: logarithm

The important rule:

\(\log(x)\) is defined only if:

\(x > 0\)

⚠️ Note: not including 0!

The difference from square root:

Square root: \(x \geq 0\) (0 allowed ✓)

Log: \(x > 0\) (0 not allowed ✗)

This is a critical difference!

Why is 0 not allowed?

\(\log(x)\) asks:
"To what power must 10 be raised?"

\(10^? = 0\)

No such power exists! ❌

\(10^{-100} = 0.000...1\) (close to 0)
But never exactly 0

Examples:

\(f(x) = \log(x)\)
Domain: \((0, \infty)\) ✓

\(f(x) = \log(x-3)\)
Domain: \((3, \infty)\) ✓

\(f(x) = \log(5-x)\)
Domain: \((-\infty, 5)\) ✓

Question 5

10.00 pts

📋 Summary of restrictions:

Which expression is the most restrictive?

Logarithm (\(x > 0\)) is the most restrictive — does not include 0

Square root is the most restrictive

Division by zero is the most restrictive

All are equal

Explanation:

📋 Summary of the 3 restrictions

Comparison:

1️⃣ Division by zero:
\(\frac{1}{x} \Rightarrow x \neq 0\)
❌ Only one point excluded

2️⃣ Square root:
\(\sqrt{x} \Rightarrow x \geq 0\)
⚠️ Half the axis excluded, 0 allowed

3️⃣ Logarithm:
\(\log(x) \Rightarrow x > 0\)
⛔ Half the axis excluded, 0 also excluded!

Order of severity:

log > sqrt > division

Logarithm is the most restrictive! 🏆

Comparison table:

Operation	Condition	Is 0 allowed?
\(\frac{1}{x}\)	\(x \neq 0\)	❌
\(\sqrt{x}\)	\(x \geq 0\)	✅
\(\log(x)\)	\(x > 0\)	❌

The golden rule:

Always check:
1. denominator ≠ 0
2. square root ≥ 0
3. log > 0

Question 6

10.00 pts

✅ Unrestricted functions:

What is the domain of \(f(x) = x^2 + 3x - 5\)?

\(\mathbb{R}\) — all real numbers

\([0, \infty)\)

\(\mathbb{R} \setminus \{0\}\)

\((0, \infty)\)

Explanation:

✅ Functions without restrictions

The rule:

If there is no:
• division
• square root
• logarithm

then the domain: \(\mathbb{R}\) ✓

"Unrestricted" functions:

1. Polynomial:
\(f(x) = x^2 + 3x - 5\)
Domain: \(\mathbb{R}\) ✓

2. Exponential:
\(f(x) = 2^x\)
Domain: \(\mathbb{R}\) ✓

3. Sine/cosine:
\(f(x) = \sin(x)\)
Domain: \(\mathbb{R}\) ✓

Why?

Any real \(x\) can be substituted!

• \(x^2\) is always defined ✓
• Addition/multiplication always work ✓
• \(2^x\) is always defined ✓

No restrictions!

More examples:

\(f(x) = x^3 - 7x + 2\)
Domain: \(\mathbb{R}\)

\(f(x) = 3^{2x} + 5\)
Domain: \(\mathbb{R}\)

\(f(x) = \cos(3x)\)
Domain: \(\mathbb{R}\)

Question 7

10.00 pts

🔍 Example:

What is the domain of \(f(x) = \frac{1}{x-2}\)?

\(\mathbb{R} \setminus \{2\}\) — all reals except \(2\)

\(\mathbb{R}\)

\(\mathbb{R} \setminus \{0\}\)

\([2, \infty)\)

Explanation:

🔍 Practical example

The function:

\(f(x) = \frac{1}{x-2}\)

The steps:

Step 1: Identify restrictions
There is division! 🚨

Step 2: Condition
Denominator \(\neq 0\)

\(x - 2 \neq 0\)

Step 3: Solve
\(x \neq 2\) ✓

Step 4: Domain
\(\mathbb{R} \setminus \{2\}\)

Explanation:

"All real numbers
except 2"

• \(x = 1.9\) ✓ allowed
• \(x = 2\) ✗ not allowed!
• \(x = 2.1\) ✓ allowed

Question 8

10.00 pts

√ Example:

What is the domain of \(f(x) = \sqrt{x+3}\)?

\([-3, \infty)\)

\([3, \infty)\)

\((-3, \infty)\)

\(\mathbb{R}\)

Explanation:

√ Example with a square root

The function:

\(f(x) = \sqrt{x+3}\)

The steps:

Step 1: Identify restrictions
There is a square root! 🚨

Step 2: Condition
The expression under the root \(\geq 0\)

\(x + 3 \geq 0\)

Step 3: Solve
\(x \geq -3\) ✓

Step 4: Domain
\([-3, \infty)\)

Explanation:

"from -3 and up"

• \(x = -4\) ✗ not allowed (\(\sqrt{-1}\))
• \(x = -3\) ✓ allowed (\(\sqrt{0}\))
• \(x = 0\) ✓ allowed (\(\sqrt{3}\))

Question 9

10.00 pts

📊 Example:

What is the domain of \(f(x) = \log(2x-6)\)?

\((3, \infty)\)

\([3, \infty)\)

\(\mathbb{R} \setminus \{3\}\)

\(\mathbb{R}\)

Explanation:

📊 Example with logarithm

The function:

\(f(x) = \log(2x-6)\)

The steps:

Step 1: Identify restrictions
There is a logarithm! 🚨

Step 2: Condition
The expression inside the log \(> 0\)

\(2x - 6 > 0\)

Step 3: Solve
\(2x > 6\)
\(x > 3\) ✓

Step 4: Domain
\((3, \infty)\)

⚠️ Note:

Open bracket! (

3 itself is not allowed!

• \(x = 3\) ✗ (\(\log(0)\) is undefined)
• \(x = 3.1\) ✓ (\(\log(0.2)\) is defined)

Question 10

10.00 pts

📚 Summary:

What is the first step in finding the domain?

Identify whether there are restrictions: division, square root, or logarithm

Solve an equation

Draw a graph

Check the range

Explanation:

📚 Summary — Domain of definition

🎯 The steps:

1️⃣ Identify restrictions:
• Any division? 🚨
• Any square root? 🚨
• Any logarithm? 🚨

2️⃣ Write conditions:
• Denominator ≠ 0
• Square root ≥ 0
• Log > 0

3️⃣ Solve:
Find which x-values satisfy

4️⃣ Write the domain:
In interval notation

The golden rule:

Restriction	Condition	Notation
Division	≠ 0	\(\mathbb{R} \setminus \{...\}\)
Square root	≥ 0	\([a, \infty)\)
Logarithm	> 0	\((a, \infty)\)

Quick examples:

\(x^2 + 5\) → \(\mathbb{R}\) (no restrictions)

\(\frac{1}{x-7}\) → \(\mathbb{R} \setminus \{7\}\)

\(\sqrt{x+2}\) → \([-2, \infty)\)

\(\log(x-1)\) → \((1, \infty)\)

⚠️ Remember:

Square root: 0 is allowed [
Log: 0 is not allowed (

This is a critical difference!