Use a piecewise linear model
A piecewise model uses a different linear rule on each stated interval. Select the interval first, then substitute the input into its rule
\[\begin{cases}y=2x+3,&0\le x\le4\\y=x+7,&4<x\le7\end{cases}\]
- Read the interval Compare the input with the interval attached to each rule before substituting \(\displaystyle 0\le x\le4\quad\text{or}\quad4<x\le7\)
- Choose one branch Use the branch whose interval contains the input \(\displaystyle x=3\Rightarrow0\le x\le4\)
- Substitute into that branch Replace x in the selected rule and evaluate the expression \(\displaystyle y=2\times3+3=9\)
- Check a shared breakpoint At a boundary, check the branch that owns the endpoint and compare adjacent values when continuity matters \(\displaystyle 2\times4+3=4+7=11\)
Worked example
A model is defined by the following rules. Find f(6)
\[\begin{cases}f(x)=3x+2,&0\le x\le2\\f(x)=x+6,&2<x\le5\\f(x)=2x+1,&5<x\le7\end{cases}\]
- \(\displaystyle 5<6\le7\)The input 6 is in the third interval, so use the third branch
- \(\displaystyle f(6)=2\times6+1\)Substitute 6 into the rule f(x)=2x+1
- \(\displaystyle 2\times6=12\)Calculate the variable term before adding the constant
- \(\displaystyle f(6)=12+1=13\)Add the constant to obtain the model value
Common mistakes
Using the wrong branch
✕\(\displaystyle f(6)=2\times6+3=15\)✓\(\displaystyle f(6)=6+7=13\)
The input 6 is in the second interval, so the second branch must be usedOverlapping intervals with different outputs
✕\(\displaystyle 0\le x\le4,\quad3<x\le7\)✓\(\displaystyle 0\le x\le4,\quad4<x\le7\)
For y=2x+3 and y=x+7, the overlap 3<x<4 gives two different outputs for one input. Use the stated intervals. A shared endpoint alone is not a problem if both rules give the same output thereAdding before multiplying in f(x)=2x+3
✕\(\displaystyle f(4)=(2+3)\times4=20\)✓\(\displaystyle f(4)=2\times4+3=11\)
In 2x+3, multiply the coefficient by x before adding the constantDifficulty levels
introEvaluate a model with two branches at inputs inside the stated intervals
standardConstruct or evaluate a two-branch model, including an endpoint or graph segment
challengeWork backwards with a three-branch model to find an input or parameter
examSelect the correct evaluation or graph interpretation for a piecewise model
Questions about this topic
Why must the interval be checked before substitution?+
The same input expression can give different values on different branches. The interval identifies which rule describes that input
How is a boundary assigned in a piecewise rule?+
The inequality signs show which branch includes the boundary. Use the branch with the inclusive inequality at that endpoint
What is the difference between a piecewise model and one linear rule?+
A single linear rule applies across one interval, while a piecewise model uses separate linear rules on separate intervals
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, General Mathematics, chapter 5: Linear functions and models
In this chapter
Substitute values into a formulaConstruct a table of values from a formulaRecognise linear equationsSolve a linear equationSet up and solve a practical linear equationBuild and use a formula in two unknownsRead coordinates and points from a straight-line graphClassify and compare gradients from graphs5D Straight-line graphs and slope: Recognise a line or find points for its graph5D Straight-line graphs and slope: Find the gradient of a line through two points5E Straight-line equations: Recognise a line or find points for its graph5E Straight-line equations: Find the gradient of a line through two pointsRead the vertical-axis intercept and gradient from an equation5E Straight-line equations: Find the equation of a line from its graph or from its intercept and slope5F Equations from graphs and points: Find the equation of a line from its graph or from its intercept and slopeFind the equation of a line through two pointsConstruct a linear model from a descriptionInterpret the graph of a linear modelRead the point of intersection of two linesSolve a pair of simultaneous linear equationsSet up a pair of simultaneous equations from a descriptionSolve a practical problem with simultaneous equationsUse a piecewise linear modelRead and construct step graphs