Functions And Modelling Exercises
Apply function models for Queensland Year 11 Mathematical Methods (QCAA). This topic brings functions to life, using function notation to describe real situations and answer practical questions without any calculus.
You will learn to evaluate and build a rule, solve for an input, state a domain and range, use piecewise functions, and find a maximum or minimum from the turning point of a quadratic — modelling skills prized across QCAA assessment.
Every question with a fully worked solution.
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Theory
In Year 11 Mathematical Methods (QCAA), a function model uses function notation \(f(x)\) to describe a real situation. This page shows how to evaluate a model, build a rule, solve \(f(x)=\text{value}\), state a sensible domain and range, and find a maximum or minimum from the vertex of a quadratic — all without calculus.
Function notation \(f(x)\) names a rule: \(f(a)\) is the output when the input is \(a\). The independent variable is the input (often time or length) and the dependent variable is the output (a cost, height or volume).
The domain is the set of allowed inputs and the range is the set of resulting outputs; in a model these are limited by what makes physical sense (you cannot have a negative time or length). A piece-wise function uses different rules on different parts of the domain, such as a flat fee for a short trip and a per-kilometre rate after that.
Evaluate a model at an input \(a\):
Maximum or minimum of a quadratic \(y=at^2+bt+c\) is at the axis of symmetry:
How to solve a modelling problem
- Set up: name the variables and write the rule \(f(x)\) from the description.
- Do the maths: evaluate \(f(a)\), solve \(f(x)=\text{value}\), or find the vertex \(t=-\dfrac{b}{2a}\) for a maximum or minimum.
- Interpret: state the answer with correct units and a sensible domain or range for the context.
Substitute \(t=4\) into the model:
| \(C(4)\) | \(=\) | \(60+90(4)\) |
| \(=\) | \(60+360\) | |
| \(=\) | \(420\) |
The cost is \(\$420\).
Set \(V(t)=300\):
| \(800-25t\) | \(=\) | \(300\) |
| \(-25t\) | \(=\) | \(300-800\) |
| \(-25t\) | \(=\) | \(-500\) |
| \(t\) | \(=\) | \(20\) |
Empty means \(V(t)=0\):
| \(800-25t\) | \(=\) | \(0\) |
| \(25t\) | \(=\) | \(800\) |
| \(t\) | \(=\) | \(32\) |
It holds \(300\) L at \(t=20\) min and is empty at \(t=32\) min.
Axis of symmetry with \(a=-5,\ b=20\):
| \(t\) | \(=\) | \(-\dfrac{b}{2a}\) |
| \(=\) | \(-\dfrac{20}{2(-5)}\) | |
| \(=\) | \(2\) |
Greatest height — substitute \(t=2\):
| \(h(2)\) | \(=\) | \(-5(2)^2+20(2)+15\) |
| \(=\) | \(-20+40+15\) | |
| \(=\) | \(35\) |
The greatest height is \(35\) m, reached at \(t=2\) s.
Build the rule — flat, then \(\$2\) per extra km beyond \(2\):
| \(C(d)\) | \(=\) | \(5,\quad 0\lt d\le 2\) |
| \(=\) | \(5+2(d-2),\quad d\gt 2\) |
For a \(5\) km trip, \(d=5\gt 2\), so use the second piece.
Substitute \(d=5\):
| \(C(5)\) | \(=\) | \(5+2(5-2)\) |
| \(=\) | \(5+2(3)\) | |
| \(=\) | \(5+6\) | |
| \(=\) | \(11\) |
The fare is \(\$11\) for the \(5\) km trip.
Common pitfalls
Frequently asked questions
What does f(a) mean in a model?
It is the output of the model when the input is \(a\); substitute \(x=a\) into the rule, for example \(C(4)=60+90(4)=420\).
How do I find a maximum without calculus?
For a quadratic \(y=at^2+bt+c\), the maximum or minimum is at the axis of symmetry \(t=-\dfrac{b}{2a}\); substitute that \(t\) back to get the value.
How do I state a sensible domain for a model?
Restrict the input to values that make physical sense — for instance \(0\le t\le 32\) for a tank that empties at \(t=32\) minutes.
What is a piece-wise function?
A function that uses different rules on different parts of the domain, such as a flat fee for a short trip and a per-kilometre rate beyond a set distance.
How do I know if a quadratic model has a maximum or a minimum?
Look at the coefficient of \(t^2\): if it is negative the parabola opens down (a maximum); if positive it opens up (a minimum).