Combining and Reversing Functions

Name: Combining and Reversing Functions
Author: StemPy Lab

Convert metres to centimetres, and then centimetres to millimetres: two functions, one after the other. Can the two steps be one? Can we go back? In this course we add, multiply and compose functions, we find when a function can be reversed, and we use functions to model real data with Python.

3 lessons, about 1 h 20 min3 simulations14 problems6 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
Algebra of functions and composition
We add, subtract, multiply and divide functions point by point, on the part of the domain that both share (and with a nonzero denominator for the quotient). The composite function (g∘f)(x) = g(f(x)) applies f first and then g. The order matters, and its domain is the set of x in the domain of f for which f(x) is in the domain of g.
1 simulation2 practice questions1 Python program5 problems1 Python lab
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Lesson 2about 30 min
One-one, onto and inverse functions
A function is one-one if different inputs always give different outputs, and onto if every element of the codomain is an output. A function that is both has an inverse f⁻¹, which undoes f. We find the inverse by swapping x and y and solving. The graphs of f and f⁻¹ are mirror images in the line y = x.
1 simulation2 practice questions1 Python program5 problems1 Python lab
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Lesson 3about 25 min
Functions as models of data
A function can describe real data. We plot the data and choose a family: a line, a parabola or a reciprocal. Then the computer finds the best numbers with numpy's polyfit, which minimises the sum of the squares of the errors. We judge the model by its typical error, and we trust it only inside the range of the data.
1 simulation2 practice questions1 Python program4 problems1 Python lab1 question set1 note
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Combining and Reversing Functions: Challenge Paper

20 questions · 50 min

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Combining and Reversing Functions

Course outline (3 lessons)

Algebra of functions and composition

One-one, onto and inverse functions

Functions as models of data

Test Your Learning

Your teacher

Discussion (0)

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