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
\[y=L+\frac{a}{\left(x+s\right)^p},\quad p\in\{1,2\},\quad a\ne0,\quad s>0,\quad x\in[0,\infty)\]
- Apply the practical domain Use only inputs allowed by the situation, even when the formula can be evaluated at other values \(\displaystyle x\in[0,\infty)\)
- Substitute exactly Put the input into the complete denominator and simplify the rational term before adding the limit value \(\displaystyle y=40+\frac{18}{4+2}=40+3=43\)
- 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 \(\displaystyle \lim_{x\to\infty}y=40,\quad y\ne40\text{ for finite }x\)
- 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
Find the modelled value at an input of 4 and its limiting value as the input increases
\[y=30+\frac{24}{x+2},\quad x\in[0,\infty)\]
- \(\displaystyle x=4\in[0,\infty)\)The chosen input is inside the practical domain
- \(\displaystyle y(4)=30+\frac{24}{4+2}=30+4=34\)Substitute the given input and simplify the rational term exactly
- \(\displaystyle \lim_{x\to\infty}\frac{24}{x+2}=0\)The rational term tends to zero as x increases without bound
- \(\displaystyle \lim_{x\to\infty}y=30,\quad y(4)=34\)The modelled value is 34 and the model approaches 30
Common mistakes
Ignoring the practical domain
✕\(\displaystyle y=40+\frac{18}{x+2},\quad D=\mathbb{R}\setminus\{-2\}\)✓\(\displaystyle y=40+\frac{18}{x+2},\quad D=[0,\infty)\)
A practical model includes only inputs allowed by the situation, even when the formula can be evaluated elsewhereTreating 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 outputIncluding 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 attainedDifficulty 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 12, VCE Units 3 & 4, Mathematical Methods, chapter 2: Functions and graphs
In this chapter
Read restricted graphs and horizontal level setsFind the gradient of a line through two pointsFind the equation of a line through two pointsSolve a pair of simultaneous linear equationsRead the vertical-axis intercept and gradient from an equationRead the point of intersection of two linesConstruct a linear model from a descriptionInterpret the graph of a linear modelTransform a quadratic and find its key featuresDetermine a quadratic rule from exact informationAnalyse signs and solve a quadratic inequalityAnalyse a cubic graph or determine its ruleInterpret and determine a transformed power functionConnect a factorised higher-degree rule and its graphSolve a higher-degree polynomial equation or inequalityApproximate solutions from intersections of two graphsClassify a relation and state its domain and rangeEvaluate and determine function rulesApply and describe transformations of a functionAnalyse the general rational-power familyFind 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 pointsWork with sets and interval notationSimplify, compare and operate with surdsCombine function rules and ordinatesRecognise equivalent polynomial formsUse a piecewise linear modelEvaluate exact circular-function values by symmetry and periodicityKey points and features of the basic sine and cosine graphsTransform and reconstruct sine and cosine functionsFeatures and transformations of tangent functionsForm and use periodic sine and cosine modelsAnalyse exponential and logarithmic graphsInterpret and use an exponential modelSelect and interpret a function modelEvaluate and interpret a restricted rational model