Learn with Amar
Teaching video
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Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Rewrite a rational function to identify asymptotes and range restrictions.
- Justify the conclusion "Vertical asymptote x=3; horizontal asymptote y=2; output two is never attained" using the stated assumptions.
Before you start
Polynomial rearrangement and reciprocal 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
Analyze f(x)=(2x+1)/(x−3).
Why this math matters
Rewrite a rational function to identify asymptotes and range restrictions. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Inputs and outputs are real.
- The numerator remainder is nonzero.
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.
Expose a rational graph's shifted reciprocal structure
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Analyze f(x)=(2x+1)/(x−3).
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.
Build the model
2x+1=2(x−3)+7
Separate a multiple of the denominator from the remainder.
Work through the mathematics
f(x)=2+7/(x−3), with x≠3
This is a shifted reciprocal graph.
Check the conclusion
Vertical asymptote x=3; horizontal asymptote y=2; output two is never attained
The nonzero reciprocal term cannot equal zero at a finite allowed input.
The result
Vertical asymptote x=3; horizontal asymptote y=2; output two is never attained
The nonzero reciprocal term cannot equal zero at a finite allowed input.
Common mistakes to catch
- A horizontal asymptote can be crossed in some other rational functions, so inspect this formula specifically.
- A denominator zero alone does not distinguish a pole from a removable hole.
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
Solve f(x)=9.
Show a hint
Set 7/(x−3)=7.
Reveal answer and explanation
x=4
The original quotient gives 9 at four.
Practice 2
Is x=3 a removable hole?
Show a hint
Check whether the numerator also vanishes.
Reveal answer and explanation
No
Its value there would be seven, so no common linear factor cancels.
Take the idea with you
Use an algebraic rewrite to separate baseline behavior from a reciprocal correction.
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: Divide complex numbers using a conjugate
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