Checking domains and the reasonableness of model predictions | Year 11

01 / LEARN

The idea, step by step

A valid manipulation is not the final check

Before solving, identify restrictions such as positive log arguments, non-zero denominators and positive exponential bases. After solving, test each candidate against these original restrictions.

A compact combined expression can have a larger domain than the original expression. An answer that solves the transformed equation may still be excluded by the starting question.

A model needs a useful contextual domain

A formula might be mathematically defined for all real t but model a process only for t between zero and an observed end time. State both the meaning and unit of a variable when interpreting a result.

Use sensible precision and practical constraints: counts are whole, money is normally rounded to cents, and a forecast cannot reasonably ignore known capacity limits. Report assumptions instead of claiming the formula proves future behaviour.

A common trap

Treating every algebraically possible input as meaningful for a practical model.

Instead: Mathematical domain and contextual validity are different checks; a fitted formula does not remove physical limits.

02 / WORKED EXAMPLES

See the thinking

Example 1

Check a log-equation candidate

Solve $log (x + 5) - log (x - 1) = log 3$ .

Both log arguments require x>1.
Combine: $(x + 5) / (x - 1) = 3$ ; hence x+5=3x-3 and x=4.
Check: $log 9 - log 3 = log 3$ , so 4 is valid.

x=4

Example 2

Separate a formula from its use

A culture model N(t)=200 times 1.4 to the power t is fitted during the first 6 hours. Its container holds at most 2000 cells. Evaluate a prediction after 10 hours.

$N (10) = 200 {(1.4)}^{10} \approx 5785.09$ .
This exceeds the stated capacity and lies outside the fitted interval.
The arithmetic is correct, but the unchanged growth model is not credible for this prediction.

About 5785 by the formula, but the forecast is not physically reasonable.

Check your understanding

Find the real domain of $log (x^{2} - 9)$ .

Show answer

$x < - 3$ or $x > 3$ .

Require $x^{2} - 9 > 0$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find the domain of $log (5 - 2 x)$ .
2 marks
Find the domain of $log x + log (x - 4)$ .
2 marks
Solve $log (x + 2) = log (3 x - 4)$ and check its domain.
3 marks
A model is $V (t) = 900 {(0.8)}^{t}$ for an asset bought at t=0. Explain why t=-2 is not a prediction of its resale value after purchase.
2 marks
A population model gives 47.826 individuals. How should a reported prediction be written, and why?
2 marks
A student solves $log x + log (x - 3) = 1$ and gives x=-2 because the combined product is 10. Explain precisely why this is invalid.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the domain of $log (5 - 2 x)$ .
2 marks
Require 5-2x>0.
$x < 2.5$
Find the domain of $log x + log (x - 4)$ .
2 marks
Both arguments must be positive.
$x > 4$
Solve $log (x + 2) = log (3 x - 4)$ and check its domain.
3 marks
Domain x>4/3; equate arguments: x+2=3x-4, so x=3.
At x=3 both arguments equal 5.
x=3
A model is $V (t) = 900 {(0.8)}^{t}$ for an asset bought at t=0. Explain why t=-2 is not a prediction of its resale value after purchase.
2 marks
The formula is mathematically defined at -2, but the stated post-purchase context uses t at least zero.
It refers to two years before purchase.
A population model gives 47.826 individuals. How should a reported prediction be written, and why?
2 marks
Individuals are counted in whole numbers. The unrounded value is useful in calculations but not as a literal count.
About 48 individuals, with the model assumptions stated.
A student solves $log x + log (x - 3) = 1$ and gives x=-2 because the combined product is 10. Explain precisely why this is invalid.
3 marks
Although their product is positive, neither original logarithm is real. Original domain x>3 excludes -2.
The original arguments -2 and -5 are not positive.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: exponential and logarithmic functions, growth models and numerical/algebraic prerequisites.

Official curriculum source ↗

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