10: Functions and formulae

Functions and Formulae
Must / Should / Could
Understand what a function is and “input” values to find the “output”values using a function / Find the “input” values given the “output” values for a function / Express a function in the form f(x)
Understand the term, “inverse function” / Find inverse functions for any given function involving one or two step operations / Express an inverse function in the form f-1(x)
Understand what a formula is and substitute values in a formula / Rearrange a formula to change the subject
Key Words: function, function machine, input, output, f(x), inverse function, f’(x), formula, formulae, substitute, rearrange, subject, variables
Starters:
Recap work on drawing maths tables to interpret equations and expressions
Show sequences and ask students to identify the rule
Activities:
Convert function machines into equations (eg where the input is x and the output is y) and vice versa
Practice rearranging equations (to make x the subject) in order to find inverse functions
Practice rearranging formulae using the maths table and using reasoning of factors
Plenaries:
Ensure all students understand that a function machine describes “what you have to do” to the input value to obtain an output value.
Students must also understand that if “y is a function of x” then the value of y depends on the value of x.
It is important that students can articulate a maths story, eg2x + 1 as “start by putting 2 x on the maths table. Get ready to get some more. 1” and that they are able to convert this to a diagram using function machines as follows; input, x × 2 + 1 output
Check students understand that a formula involves 2 or more variables, or unknowns.
Ensure students can draw maths tables to describe a formula.
Learning Framework Questions:
  • What does it mean if y is a function of x?
  • How can we use inverse function machines? [ie to find the input given the output value]
  • What is the difference between an equation, an expression and a formula?
  • What does variable mean?

Resources:
Maths table / resources table Cards (x’s, 1’s and their negative equivalents)
Place mats with dual maths tables / resources table MMMS worksheets
Possible Homeworks:
Convert function machines to functions / equations and vice versa
Teaching Methods/Points:
Functions and function machines
A function expresses a dependence between two variables (i.e. one variable is independent while the other variable is dependent on the first.) As such, the topic of functions is really a consolidation of the work already covered on algebraic equations. Indeed, the topic of functions is best explained as being no different from the work already covered on equations.
If “y is a function of x” it means that “y is dependent on x” and this is written as y = f(x).
Basically as the variable x changes [the independent variable], the variable y will change [the dependent variable]. This explains the terms, “input” and “output”;
The input is the value which we substitute for x [the independent variable].
The output is the value which we substitute for y [the dependent variable].
This has an important significance for students when using equations to plot points on a graph …
eg y = 2x + 1 … y is a function of x [y = f(x)] because it is dependent on the value of x.
To find the value of y using the maths table; “start by putting 2 times the value of x on the maths table. Get ready to get some more. 1. Count and see how much there is. This is equal to the value of y.” Therefore, by substituting values for x, the value of y can be derived, being dependent on the value of x.
Using function machines to describe dependency between two variables
So, if y is a function of x, y is dependent on x, meaning that you have to do something to x to find the value of y.
This can be described as an equation; eg y = 2x + 1
or a function; eg f(x) = 2x + 1
or as function machines; eg input output

The important point to stress with students is that all of these express exactly the same thing. They all suggest that the output (y) is found by doing things [called operations!] to the input (x).
Students, therefore, need to practice manipulating function machines in the following ways;

  • Use function machines to find the output, given the input eg 3 ……..?
for one or more operations
  • Look at inputs and outputs and describe the function machine eg 2 5
or function machines for more than one operation

  • Use function machines to find the input, given the output eg …. ? 6
for one or more operations
and they need to practice expressing equations as functions and function machines and vice versa. In order to do this, students need to practice reading the maths story and acting out the real story.
For example, given the following function machines;

Input (x) Output (y)
this is read as, “start by putting x on the maths table. Love it so much you do it two times in total. Get ready to remove some. 1. [can’t do it, add a zero to the maths table!].”
≡ 2x + -1

This can be expressed as a function; f(x) = 2x + -1 or as an equation; y = 2x + -1
Equally, an equation can be described using function machines. Take, for example, the equation, y = 3x + 2;
this is read as, “start by putting x on the maths table 3 times. Get ready to get some more. 2.”

Input (x) Output (y)

Finding inverse functions
The inverse function, f-1(x) changes the dependency of the variables x and y.
In terms of function machines, it involves finding inverse operations in order to reverse the process and start with the output value (y) and derive the input value (x) – although by the nature of the change of dependency, y actually becomes the input value!
In terms of equations, it involves rearranging to change the subject of the equation from y to x. This also explains how the process of finding inverse operations is equivalent.
For example;

Input (x) Output (y) is equivalent to y = 2x + -1



To make x the subject … remove negative 1 from both sides



Now; y + 1 = 2x using reasoning of factors if (y + 1) = 2 × x then x =
and this can be translated as a function machine diagram as follows;
Output Input which is written as an inverse function; f-1(x)=


Formulae: substituting and rearranging
A formulasimply describes the relationship between two or more variables. Therefore, an equation and a function are types of formulae. Generally speaking there will be one dependent variable and one or more independent variables in a formula; egf = ma … f (force) is dependent on inputting the values for m (mass) and a (acceleration), which are the independent variables in this example.
Students need to be able to substitute values into a formula to calculate the value of a specific variable, and they need to be able to rearrange formulae, using the maths table to help them.
Substituting values
Generally, reading the maths story and acting out the maths story as a real story using the maths table will help students understand the process of substituting. However, it is essential that students are also able to apply reasoning to terms involving products, egma should be interpreted as m × a.
Example:
Find the value of p when q is 2 and r is 3 given the formula; p = qr + r – 1


By substituting the values for the letters the maths table looks like this:
qr≡ q × r ≡ 2 × 3 = 6


therefore: p = 6 + 3 + -1 = 8
Rearranging formulae
Rearranging a formula changes the subject of a formula. This process is used to find inverse functions, as described on the previous page.
Example rearrange the formula f = 2a + 1 to make a the subject


Add negative 1 to both sides to eliminate the 1.


So; f = 2a … this can be rearranged using reasoning of factors … f + -1 = 2 × a so (f + -1) ÷ 2 = a
Therefore
Functions and FormulaeHelp Sheet
Functions
A function describes a relationship between two variables. So, if y is a function of x, y is dependent on x, meaning that you have to do something to x to find the value of y. This is written as y = f(x).
This makes x the value that you input and y isthe output value.
This can be described as an equation; eg y = 2x + 1
or a function; eg f(x) = 2x + 1
or as function machines; eg input (x) output (y)
For the equation, function, and function machines shown, the value of y, or “f(x)” is equal to;


So,
  • if the value of the input (x) is 1, the value of the output (y) is 3.
  • if the value of the input (x) is 2, the value of the output (y) is 5.
  • if the value of the input (x) is 3, the value of the output (y) is 7. …
Inverse functions
The inverse of a function, f(x) is written as f-1(x).
The inverse function machines can be found by finding inverse operations as follows;
input (x) output (y)

output (y) input (x)
This can be used to find input values if you are given the output values.
For an inverse function, the output becomes the input and the input becomes the output, as shown.
So, the inverse function is written as: f-1(x) =
Formulae
A formulasimply describes the relationship between two or more variables. Therefore, an equation and a function are types of formulae. The value of a variable in a formula can be found by substituting values for the other variables.
Eg a = if b = 2 and c = 3 ... a = = = 3
Formulae can be rearranged using maths tables.
Example rearrange the formula f = 2a + 1 to make a the subject




So; f + -1= 2a … this can be rearranged using reasoning of factors … f + -1 = 2 × a so (f + -1) ÷ 2 = a
Therefore

10: Functions and formulae