DPST1012 Chap.4 Function Models, Equations and Inequalities
Function Models, Equations and Inequalities
Three families of models
A worksheet in the functions topic introduces the three model families the course returns to all term. Linear functions f(x) = ax + b describe change at a constant rate, such as average height against age through childhood. Trigonometric functions describe oscillation, such as the height of a tide through a day.
Exponential functions describe rapid growth, such as Australia's population over the last century. Each model is fitted to data, evaluated at a point, compared with the data, and then questioned for its limitations.
Two questions: f(x) = c and f(x) > c
The course names two ideas that run underneath everything: solving f(x) = c and solving f(x) > c.
From a table, solving means listing every input whose output meets the condition. From a continuous graph, it means drawing the horizontal line y = c and reading where the curve meets it or lies above it.
Removing the story from a problem so that only inputs and outputs remain is called abstraction, and it lets the same method work for unemployment rates, temperatures or a rollercoaster track.
Reading models critically
A model is useful inside the data it was built from and doubtful outside it.
Estimating between two data points is interpolation; projecting beyond the last point is extrapolation, and it assumes the pattern continues. The difference between a data value and the model's value at the same input shows where a model bends too early or too late. Strict and non-strict inequality signs decide which borderline values belong in an answer.
What this chapter covers
- 01
Linear, trigonometric and exponential model forms
- 02
Evaluating a fitted model and its error
- 03
Limitations of a model outside its data
- 04
Solving f(x) = c from a table
- 05
Solving f(x) > c and double inequalities
- 06
Solving from a continuous graph
- 07
Counting solutions with a horizontal line
Worked example · free
Solve equations and inequalities from a table of values
- 1S(h) = 260 at h = 12 and h = 13, so the solution set is {12, 13}.
- 2S(h) > 180 for outputs 210, 260, 260 and 190: h ∈ {11, 12, 13, 14}; 180 itself is excluded.
- 1150 ≤ S(h) < 210 for outputs 180, 150 and 190: h ∈ {9, 10, 14}.
- 1Two inputs share the output 260, so S is not one-to-one.
Key terms
- Mathematical Model
- An equation or function chosen to represent a real system so that its behaviour can be predicted.
- Extrapolation
- Estimating a value beyond the range of the data, which assumes the observed pattern continues.
- Abstraction
- The step of removing a problem's context so that only the inputs, outputs and their relationship remain.
- Residual
- The difference between an observed data value and the value a model predicts at the same input.
- Exponential Model
- A function of the form a·e to the power bx plus c, used for quantities that grow or decay rapidly.
Function Models, Equations and Inequalities FAQ
Why do so many questions ask me to solve f(x) = c?
Because most real questions take that form once the context is removed: when does the temperature reach 20 degrees, which months have a given rate, how long until a tank is empty. The course treats it as a core idea.
How should I write the answer when solving from a table?
As a set of the inputs that work, such as {Feb, Mar, Apr}, because a table only contains the listed values. Intervals belong to continuous graphs, where every value between two points is included.
Is extrapolating a model always wrong?
No, but it is riskier than interpolating, because it assumes the pattern holds where there is no data. State the assumption in words, and expect projections far beyond the data to be unreliable.
What counts as a limitation of a fitted model?
A real reason the pattern could fail, such as growth stopping in adulthood, tides changing over the month, or population growth slowing. Saying only that the model is not exact usually earns little credit.
What is the difference between a strict and a non-strict inequality in an answer?
A strict sign, less than or greater than, excludes the boundary value, while a non-strict sign includes it. In a table this decides whether a borderline month or hour is listed; on a graph it decides between an open and a filled endpoint.
Exam move
Take one data set, fit a straight line by eye, compute two residuals, and write one sentence on where the model fails; then solve two equations and one inequality from the same table. Note every boundary value you had to decide on, and why it was in or out.
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