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Evaluate and interpret a restricted rational model

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For the displayed model, calculate the denominator when the input is 4

\[y=40+\frac{18}{x+2},\quad x+2=4+2=?\]
Year 11, VCE Units 1 & 2, Chapter 6, 4 difficulty levels, Calculator allowedBuild a worksheet

Interpreting a restricted rational model

A restricted rational model is evaluated only on its practical domain. Its range is found from the boundary output and the horizontal limit, remembering that the limit may be approached without being attained

\[y=L+\frac{a}{\left(x+s\right)^p},\quad p\in\{1,2\}\]
  1. Apply the practical domain Use only inputs allowed by the situation, even when the formula can be evaluated at other values \(\displaystyle x\in[0,\infty)\)
  2. Substitute exactly Put the input into the complete denominator and simplify the rational term before adding the limit value \(\displaystyle y=40+\frac{18}{4+2}=40+3=43\)
  3. Separate limit from value The horizontal limit describes the value approached as the input grows; it is not reached at a finite input when the rational term is non-zero \(\displaystyle \lim_{x\to\infty}y=40,\quad y\ne40\text{ for finite }x\)
  4. Use the boundary for range Find the output at the included domain boundary and combine it with the unattained horizontal limit \(\displaystyle y(0)=40+\frac{18}{0+2}=49,\quad y\in(40,49]\)

Worked example

For the model shown, find the modelled value when the input is 4 and its limiting value as the input increases

\[y=30+\frac{24}{x+2},\quad x\in[0,\infty)\]
  1. \(\displaystyle x=4\in[0,\infty)\)The chosen input is inside the practical domain
  2. \(\displaystyle y(4)=30+\frac{24}{4+2}=30+4=34\)Substitute x=4 and simplify the rational term exactly
  3. \(\displaystyle \lim_{x\to\infty}\frac{24}{x+2}=0\)The rational term tends to zero as x increases without bound
  4. \(\displaystyle \lim_{x\to\infty}y=30,\quad y(4)=34\)The modelled value is 34 and the model approaches 30

Common mistakes

Ignoring the practical domain
✕\(\displaystyle y=40+\frac{18}{x+2},\quad D=\mathbb{R}\)✓\(\displaystyle y=40+\frac{18}{x+2},\quad D=[0,\infty)\)
A practical model includes only inputs allowed by the situation, even when the formula can be evaluated elsewhere
Treating the limit as attained
✕\(\displaystyle \lim_{x\to\infty}y=40\quad\Rightarrow\quad y=40\text{ at a finite }x\)✓\(\displaystyle \lim_{x\to\infty}y=40\quad\text{and}\quad y\ne40\text{ for finite }x\)
The rational term approaches zero only as x increases without bound, so the limit is not a finite-input output
Including the limit in the range
✕\(\displaystyle y=40+\frac{24}{x+1},\quad x\in[0,\infty)\quad\Rightarrow\quad y\in[40,64]\)✓\(\displaystyle y=40+\frac{24}{x+1},\quad x\in[0,\infty)\quad\Rightarrow\quad y\in(40,64]\)
The boundary value 64 is included, but the horizontal limit 40 is approached and not attained

Difficulty levels

introEvaluate a model at an allowed input and identify the practical domain
standardInterpret the direction, limiting value and range of a restricted rational model
challengeSolve for an input or compare modelled levels using exact rational values
examCombine domain, substitution, limit and range reasoning in a model interpretation

Questions about this topic

What is a practical domain?+

It is the set of input values allowed by the situation being modelled. It can be narrower than the algebraic domain of the formula

Why is a horizontal limit often not in the range?+

The rational term is non-zero at every finite allowed input, so the model approaches the limit without reaching it

How are the range endpoints found?+

Evaluate the model at an included domain boundary and compare that output with the limiting value, noting which endpoint is attained

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 6: Functions and relations

In this chapter

Classify a relation and state its domain and rangeEvaluate and determine function rulesFind the maximal domain of an algebraic functionAnalyse reciprocal and square-reciprocal functionsAnalyse sideways-parabola and root-function familiesDetermine a power-family rule from features and pointsDetermine a circle's equation and featuresAnalyse semicircles and circular regionsApply and describe transformations of a functionIntersections and tangents of a line and a circleEvaluate and interpret a restricted rational modelForm and analyse an inverse relation or function