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Teaching video
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Grade 9 chapters and video availability01 · Read and understand
What you will learn
- Intersect denominator and real-square-root restrictions.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Signed arithmetic, fraction and decimal operations, one-step equations, and introductory coordinate graphs.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Find the real domain of f(x)=√(x−2)/(x−5).
Why this math matters
List algebraic and contextual input restrictions before making a graph or solving an equation. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use the real-number system and the restrictions or data model stated in the question unless another domain is specified.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Find inputs excluded by an expression
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find the real domain of f(x)=√(x−2)/(x−5).
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the conditions
x−2 ≥ 0 gives x ≥ 2
A real square root requires a nonnegative radicand.
Develop the calculation
x−5 ≠ 0 excludes x=5
Division by zero is not defined.
Check and interpret
Domain = [2,5) ∪ (5,∞)
An input must satisfy both restrictions; two is allowed because a zero numerator is valid.
The result
Domain = [2,5) ∪ (5,∞)
An input must satisfy both restrictions; two is allowed because a zero numerator is valid.
Common mistakes to catch
- A zero numerator is allowed when the denominator is nonzero.
- Domain conditions apply together rather than as alternatives.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Find the real domain of 1/(x+3).
Show a hint
Exclude the denominator's zero.
Reveal answer and explanation
All real numbers except −3
No square-root or other restriction adds another excluded region.
Practice 2
Is x=2 allowed in the main function?
Show a hint
Evaluate numerator and denominator separately.
Reveal answer and explanation
Yes; f(2)=0
The numerator is zero but the denominator is −3, so the quotient is defined.
Take the idea with you
List algebraic and contextual input restrictions before making a graph or solving an equation.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Follow the order of two function machines
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