Reading a restricted graph
Read the domain from the allowed horizontal inputs and the range from the vertical outputs produced on those inputs. Include the whole restricted graph, including interior turning points and endpoint symbols. For the example below, let f(x) = (x − 1)² for −2 ≤ x ≤ 4.
\[f(x)=k\quad\Longleftrightarrow\quad\text{the graph meets }y=k\]
- Read domain and range on their axes The domain is the set of allowed x-values and the range is the set of resulting y-values \(\displaystyle x\in\left[-2,4\right],\quad y\in\left[0,9\right]\)
- Honour endpoint symbols A square bracket includes an endpoint and a round bracket excludes it in a domain or range interval \(\displaystyle \left(-2,4\right]\text{ excludes }-2\text{ and includes }4\)
- Use the whole restricted graph An interior turning point can determine an extreme value, so endpoint heights alone may not give the range \(\displaystyle f(-2)=9,\quad f(1)=0,\quad f(4)=9\quad\Rightarrow\quad\operatorname{range}(f)=\left[0,9\right]\)
- Use horizontal levels Solutions of f(x) = k are intersections with the horizontal line y = k, and an inequality uses the x-values on the required side \(\displaystyle f(x)=k\quad\Longleftrightarrow\quad\text{intersections with }y=k\)
Worked example
Read the key features of the restricted function below, including its domain, range, x-intercept and minimum
\[f(x)=\left(x-2\right)^{2},\quad -3\le x\le5\]
- \(\displaystyle D_f=\left[-3,5\right]\)The stated closed interval is the domain
- \(\displaystyle f(2)=\left(2-2\right)^2=0\quad\Rightarrow\quad\text{minimum }\left(2,0\right)\)The turning point lies inside the domain and gives the minimum
- \(\displaystyle f(-3)=\left(-3-2\right)^2=25,\quad f(5)=\left(5-2\right)^2=9\)Evaluate both endpoints to find the largest output on the restricted interval
- \(\displaystyle \operatorname{range}(f)=\left[0,25\right],\quad\text{x-intercept}=\left(2,0\right)\)The minimum and the larger endpoint value give the range; the graph meets the horizontal axis at the stated intercept
Common mistakes
Using the plotting window as the domain
✕\(\displaystyle \text{plotting window }\left[-5,5\right]\Rightarrow D_f=\left[-5,5\right]\)✓\(\displaystyle \text{stated restriction }\left[-2,4\right]\Rightarrow D_f=\left[-2,4\right]\)
The plotting window is only a display choice; the stated restricted interval gives the function's domainUsing endpoint heights only for the range
✕\(\displaystyle f(-2)=f(4)=9\quad\Rightarrow\quad\operatorname{range}(f)=\left[9,9\right]\)✓\(\displaystyle f(1)=0,\quad f(-2)=f(4)=9\quad\Rightarrow\quad\operatorname{range}(f)=\left[0,9\right]\)
An interior minimum can determine the range, so the two endpoint heights are not enoughCounting turning points for a level equation
✕\(\displaystyle f(x)=k\quad\Rightarrow\quad\text{number of solutions}=\text{number of turning points}\)✓\(\displaystyle f(x)=k\quad\Rightarrow\quad\text{number of solutions}=\text{number of intersections with }y=k\)
Solutions come from intersections with the horizontal level, not necessarily from turning pointsDifficulty levels
introRead the domain and range of a simple restricted polynomial
standardAdd intercepts, endpoint inclusion and a count of solutions to a level equation
challengeSolve a level inequality or partition k by the number of level-set solutions
examUse a mixed graph representation with a feature choice and one level-solution count
Questions about this topic
What is the difference between domain and range?+
The domain is the set of allowed input values x, while the range is the set of output values f(x) produced by those inputs
Why does f(x) = k use a horizontal line?+
The equation asks where the output equals the chosen level, so its solutions are the input coordinates where the graph intersects that horizontal line
Why does the plotting window not set the domain?+
A plotting window controls what is visible on the display. The domain is determined by the interval or endpoint information stated for the function
Can I use a calculator?+
No. This topic is practised without a 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