Academic Skills Advice
Functions
Function notation is another way of writing equations.
For example: instead of writing 𝒚 = 𝟕𝒙 + 𝟑, we could write 𝒇(𝒙) = 𝟕𝒙 + 𝟑
(See lesson 2 for more information about function notation).
𝑥
is the INPUT and is called the DOMAIN.
𝑓(𝑥)
is the OUTPUT and is called the RANGE.
A function should only have one 𝒚-value for every 𝒙-value.
𝒇(𝒙) = 𝒙𝟐 + 𝟑 is a function because when you put a value in there is only one
possible answer.
𝒈(𝒙) = √𝒙
is not a function as each value you put in has 2 possible answers (e.g.
√9 = ±3).
Evaluating functions with numbers:
We looked at evaluating functions in lesson 2, but here is a recap: If you are asked to
evaluate a function you need to put in the 𝑥-value (domain) and find the resulting 𝑦-value
(range).
Examples:

Given the function: 𝒇(𝒙) = −𝒙 + 𝟒
Find 𝑓(8)
This just means wherever you see an 𝑥 replace it with an 8.
𝑓(𝑥) = −𝑥 + 4
𝑓(8) = −(8) + 4
𝑓(8) = −4

Given the function: 𝒈(𝒙) = 𝟒𝒙𝟐 − 𝟖𝒙 + 𝟐
Find 𝑔(3)
Now wherever you see an 𝑥 replace it with a 3.
𝑔(𝑥) = 4𝑥 2 − 8𝑥 + 2
𝑔(3) = 4(3)2 − 8(3) + 2
𝑔(3) = 36 − 24 + 2
𝑔(3) = 14
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Evaluating functions with Algebra:
Whatever you are asked to find the function of just replace the 𝑥 with it, no matter how
complicated.
Examples:

Given the function: 𝒉(𝒙) = 𝟑𝒙 − 𝟕
Find 𝒉(𝒙 + 𝟐)
Wherever you see an 𝑥 replace it with (𝑥 + 2).
ℎ(𝑥) = 3𝑥 − 7
Replace 𝑥 with (𝑥 + 2):
Tidy up:

ℎ(𝑥 + 2) = 3(𝑥 + 2) − 7
= 3𝑥 + 6 − 7
= 3𝑥 − 1
Given the function: 𝒇(𝒙) = 𝒙𝟐 + 𝟐𝒙 − 𝟒
Find 𝒇(𝒙 − 𝟑)
Wherever you see an 𝑥 replace it with (𝑥 − 3).
𝑓(𝑥) = 𝑥 2 + 2𝑥 − 4
Replace 𝑥 with (𝑥 − 3):
Multiply all brackets out:
Tidy up:

𝑓(𝑥 − 3) = (𝑥 − 3)2 + 2(𝑥 − 3) − 4
= 𝑥 2 − 6𝑥 + 9 + 2𝑥 − 6 − 4
= 𝑥 2 − 4𝑥 − 1
Given the function: 𝒈(𝒙) = −𝟒𝒙 + 𝟑
Find 𝒈(𝒙𝟐 − 𝒙 + 𝟐)
Wherever you see an 𝑥 replace it with (𝑥 2 − 𝑥 + 2).
𝑔(𝑥) = −4𝑥 + 3
Replace x with (x 2 − x + 2):
Multiply out bracket:
Tidy up:
© H Jackson 2012 / ACADEMIC SKILLS
𝑔(𝑥 2 − 𝑥 + 2) = −4(𝑥 2 − 𝑥 + 2) + 3
= −4𝑥 2 + 4𝑥 − 8 + 3
= −4𝑥 2 + 4𝑥 − 5
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Composite Functions:
You may be asked to combine 2 functions. This can be written as 𝑓𝑔(𝑥) or 𝑓(𝑔(𝑥)). To
work it out you apply 𝑔 (the inside function) first then evaluate 𝑓. If you were asked to do
𝑔𝑓(𝑥) you would apply 𝑓 (the inside function) first then evaluate 𝑔.
Examples:

Given the functions: 𝒇(𝒙) = 𝒙𝟐 + 𝟐
𝒇𝒈(𝟐)
Find:
𝒈𝒇(𝟒)
𝒈𝟐 (𝟐)
𝒇𝒈(𝟐)
Find 𝑔(2):
Find 𝑓 (𝑎𝑛𝑠):
𝑔(2) = 2(2) − 1 = 3
𝑓 (3) = 32 + 2 =
11
𝒈𝒇(𝟒)
Find 𝑓(4):
Find 𝑔(𝑎𝑛𝑠):
𝑓(4) = 42 + 2 = 18
𝑔(18) = 2(18) − 1 = 35
and
𝒈(𝒙) = 𝟐𝒙 − 𝟏
𝒇𝒈(𝒙)
𝒈𝟐 (𝟐) this is the same as doing 𝑔𝑔(2)
Find 𝑔(2):
𝑔(2) = 2(2) − 1 = 3
Find 𝑔(𝑎𝑛𝑠):
𝑔(3) = 2(3) − 1 = 5
𝒇𝒈(𝒙)
Find 𝑔(𝑥):
Find 𝑓(𝑎𝑛𝑠):
𝑔(𝑥) = 2𝑥 − 1
𝑓(2𝑥 − 1) = (2𝑥 − 1)2 + 2
= 4𝑥 2 − 4𝑥 + 1 + 2
= 𝟒𝒙𝟐 − 𝟒𝒙 + 𝟑
Inverse Functions:
If you have a function 𝑓(𝑥) then the notation for the inverse function is 𝒇−𝟏 (𝒙). You find an
inverse function by “undoing” the original function. So if the original ended up with “add 3”
then the inverse would start with “subtract 3” etc.
This is the method to find the inverse:




Replace the function notation with 𝑦 =
Rearrange the function to make 𝑥 the subject
Replace 𝑥 with 𝑦 and vice versa.
Put the function notation back in.
(As we are undoing the original function no matter what 𝑓(𝑥) is 𝑓 −1 (𝑓(𝑥)) = 𝑥.)
Work through the following examples for more guidance on finding inverse functions.
© H Jackson 2012 / ACADEMIC SKILLS
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Examples:

Given the function: 𝒇(𝒙) = 𝟕𝒙 − 𝟒, find 𝒇−𝟏 (𝒙)
Replace the function notation with y =:
𝒚 = 𝟕𝒙 − 𝟒
Rearrange the function to make x the subject:
𝒙= 𝟕+𝟕
𝒚
𝟒
(See Lesson 3 for help with rearranging)
𝒙
𝟒
Replace x with y and y with x:
𝒚=𝟕+𝟕
Put the function notation back in:
𝒇−𝟏 (𝒙) = 𝟕 + 𝟕

𝒙
𝟒
Given the function: 𝒈(𝒙) = 𝒙𝟐 + 𝟑, find 𝒈−𝟏 (𝒙)
Replace the function notation with y =:
𝒚 = 𝒙𝟐 + 𝟑
Rearrange the function to make x the subject:
𝒙 = √𝒚 − 𝟑
Replace x with y and y with x:
𝒚 = √𝒙 − 𝟑
Put the function notation back in:
𝒈−𝟏 (𝒙) = √𝒙 − 𝟑
Asymptotes:
Some functions have certain values of 𝑥 which cannot be used.
For example: 𝒇(𝒙) =
𝟒
𝒙−𝟐
In this function 𝒙 cannot equal 2 because that would
make the bottom of the fraction 0 and you cannot
divide by 0.
If you were to plot this function on a graph then there would be a line at 𝑥 = 2 which the
graph cannot cross, this line is known as an asymptote.
Asymptote
© H Jackson 2012 / ACADEMIC SKILLS
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Finding the Domain and the Range:
The idea of asymptotes can be used to help to find the domain of a function.
Domain
The domain is all the values that can go into the function.
Probably the easiest way to find the domain is to look for any values that can’t go in.
In the example above the domain would be: all real numbers except 2 (i.e. 𝒙 ∈ ℝ, 𝒙 ≠ 𝟐)
The main things to look for are:
can’t divide by 0
can’t square root a negative number
𝑥 is a member of
the real numbers.
Examples:

𝟑
Find the domain of the function 𝒇(𝒙) = 𝒙+𝟓
(think of any values that cannot be put into the function)
Domain:

𝑥 ∈ ℝ, 𝑥 ≠ −5
(𝑥 can be anything except -5)
Find the domain of the function 𝒈(𝒙) = √𝒙 − 𝟑
Domain:
𝑥 ∈ ℝ, 𝑥 ≥ 3
(𝑥 has to be 3 or above so that the number in
the square root sign isn’t negative)
Range
The range is all the values that can come out of the function.
To find the range just use common sense and think if there is anything that can’t come out
of the function. Also a function should only have one possible answer so you need to
limit the range to validate the function.
Examples:

𝟑
Find the range of the function 𝒇(𝒙) = 𝒙+𝟓
(think of any values that cannot come out of the function)
Range:
𝑓(𝑥) ∈ ℝ, 𝑓(𝑥) ≠ 0 (not possible to divide 3 and end up with 0)
N.b. if the 𝑥 is on the top then we
could get 0 as 0÷anything = 0.

Find the range of the function 𝒈(𝒙) = √𝒙 − 𝟑
Range:
© H Jackson 2012 / ACADEMIC SKILLS
𝑔(𝑥) ≥ 0
(By specifying that the answer has to be
greater than 0 we have validated the function
as there is now only 1 possible answer for
each 𝑥 value)
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