FM-Functions-Piecewise-Functions (1).pptx

Piecewise Functions
In mathematics, a piecewise-defined function (also called
a piecewise function or a hybrid function) is a function defined by
multiple sub-functions, each sub-function applying to a certain
interval of the main function's domain, a sub-domain.

Piecewise Functions
Sometimes functions are defined in ‘pieces’, with a different function for different
ranges of values.

Piecewise Functions
(-1, 0)
(2, 9)
(5, 0)
(0, 5)
Sketch >
Sketch >
Sketch >

Piecewise Functions
Sketch >
Sketch >
Sketch >
𝑓 (𝑥 )=
{
𝑥
2
0 ≤ 𝑥<1
1 1 ≤ 𝑥 <2
3 − 𝑥 2 ≤ 𝑥 <3
(1, 1) (2, 1)
(3, 0)
This example
was used on the
specification
itself!

Range of Piecewise Functions
The sketch shows the graph of with the domain
Determine the range of .
Range:
Graph ? Range ?

Test Your Understanding
The function is defined for all :
Determine the range of .
Range:
Graph ? Range ?

FM-Functions-Piecewise-Functions (1).pptx

Exercise
Work out the range for each of these
functions.
(a) for all
(b)
(c)
(a)
Give a reason why is not a suitable
domain for .
It would include 3, for which is
undefined.
(b) Give a possible domain for
The range of is
Work out and .
[Set 1 Paper 2] (a) The function is defined
as:
The range of is
Work out the value of .
(b) The function is defined as
for all .
(i) Express in the form
(ii) Hence write down the range of .
[June 2012 Paper 1] for all values of .
(a) What is the value of ?
(b) What is the range of ?
1
2
3
4
5
?
?
?
?
?
?
?
?
?
?
?

Exercise
[Jan 2013 Paper 2]
(a) What is the range of ?
(b) You are given that .
Work out the value of .
By completing the square or otherwise,
determine the range of the following
functions:
(a) for all
Range:
(b) for all
Range:
6
7
8
Here is a sketch of
for all , where is a constant. The
range of is . Work out the value
of .
?
?
?
?
?

Exercise
The straight line shows a sketch of for
the full domain of the function.
(a) State the domain of the function.
(b) Work out the equation of the line.
is a quadratic function with domain all real
values of . Part of the graph of is shown.
(a) Write down the range of .
(b) Use the graph to find solutions of the
equation .
(c) Use the graph to solve .
9 10
?
?
?
?
?

Exercise
The function is defined as:
Work out the range of .
The function has the domain
and is defined as:
Work out the range of .
[June 2012 Paper 2] A sketch of for
domain is shown.
The graph is symmetrical about . The
range of is .
Work out the function .
11 13
12
?
?
?

Constructing a function
June 2013 Paper 2
𝑥
𝑦
1 5
3
11
What would be the simplest
function to use that has this
domain/range?
A straight line! Note, that
could either be going up or
down (provided it starts and
ends at a corner)
What is the equation of this?
?
?

Constructing a function
Sometimes there’s the additional constraint that the function is ‘increasing’ or
‘decreasing’.
is a decreasing function with domain and range .
𝑦
4 6
7
19
𝑥
Graph
Function

Exercise
Domain is . Range . is an increasing function.
Domain is . Range . is a decreasing function.
Domain is . Range . is an increasing function.
1
2
3
4
5
?
?
?
?
?

FM-Functions-Piecewise-Functions (1).pptx

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