# C3 Cheat Sheet

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• Created on: 11-07-15 23:38

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C3 Cheat Sheet
Chapter Usual types of questions Tips What can go ugly
1 ­ Algebraic Almost always adding or subtracting Factorise everything in each fraction first. e.g. If denominators Blindly multiplying the two denominators when
Fractions fractions. and , common denominator will be there might be a common factor, e.g.
Simplifying top heavy fraction using and should become
algebraic division. Otherwise, just a case of practice! rather than
Simplifying fractions by first If adding/subtracting a constant, turn into a fraction. .
factorising numerator and Classic sign errors when subtracting a fraction.
denominator, where possible.
Suppose we were asked to turn into a mixed number. By
algebraic division we find the quotient is and the remainder Note use of brackets around ensures -2
becomes +2.
2. Remember we can express the result as where
is the quotient, the remainder and the divisor, just as
where 5 was the quotient and 1 the remainder.
Thus
In general, to simplify `fractions within fractions', multiply top and
bottom of the outer fraction by the denominator of the inner
fraction, e.g.:
2 ­ Functions Specifying range or domain of You should know and understand why Having a lack of care in domains/ranges with the
function. I avoid getting domain and range mixed up by thinking "ddrrr" with strictness/nonstrictness of the bound. For
Finding inverse of function. a silly voice. i.e. Domain first (possible inputs) followed by Range , range is not .
Finding specific output of function, (possible outputs) Similarly range for quadratics are non-strict
e.g. or Learn the domains and ranges of each of the `common functions' because min/max point is included.
Finding specific output using graph ( , , , ...) Putting the range of a function in terms of
only (without explicit definition of instead of say the correct . Similarly for the
Domains can be restricted for 2 main reasons:
function) range of an inverse, if the domain of the original
o The denominator of a fraction can't be 0, so domain of
Finding composite function. function was say , then the range of the
is . Use symbol.
Sketching original function and inverse is (i.e. even though the
inverse function on same axis (i.e. o You can't square root a negative number, so range of inequality is effectively the same, we're
reflection in ) is . Similarly you can't log 0 or negative referring to the output of now so need to
Be able to find or for numbers so domain (notice it's strict) use rather than )
example, when the function is Range can be restricted for 3 main reasons: When finding , accidentally doing first
known. o The domain was restricted. e.g. If and domain is and then .
"If find all values of set to , then range is . Notice If , then you should recognise that
for which " strictness/non-strictness of bounds. , NOT , which would
o Asymptotes. For reciprocals a division can never yield 0 suggest you don't quite fully understand how
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### Page 2

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Use sketch! For exponential functions, output is
(strictly) positive:
o Min/max value of a quadratic (or any polynomial whose
highest power is even). Note bound is non-strict as
min/max value included.
Range of is
For composite functions, if given say , write as ( ) then
substitute with its definition so you have
. You're less likely to go wrong.
Remember that domain is specified in terms of and the range in
terms of (or or otherwise).…read more

### Page 3

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Some questions are `quadratics in disguise':
Since , this suggest we multiply everything by , which
gives us: . Rearranging:
Then we could make the substitution and factorise, or just
factorise immediately to get
Thus , thus
If you'd managed to factorise an expression to say ,
remember that can't be 0 (as the range of exponential functions
is )
Suppose the population is given by where
is the number of years.…read more

### Page 4

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­ Transforming Using an existing function The way to remember | | vs | | is remember that changes Forgetting to add key coordinates to your
graphs of to sketch | | and/or inside the function brackets affects the values and changes diagrams, e.g. intercepts with the axis, turning
functions | | outside affect the values. Thus in | | , any negative is points, etc.
As at GCSE/C1, be able to sketch a made positive before being inputted into the function.…read more

### Page 5

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As with C2 questions, when you have a combination of say
and , then change the squared term using the identity, so that
you end up with a quadratic equation in terms of one trig function.
As per C2, if you're given a range for your solutions, then rewrite
the range as appropriate. E.g. If and you had
, then . This ensures you don't lose solutions.…read more

### Page 6

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You will often have to simplify a differentiated expression. negatives floating about), you are advised to
Remember that when factorising things with indices, factor out the work out and separately first rather than
smallest power, and factor out any fraction using the lowest write all in one go.
common multiple:
( )
If the "tangent is parallel to the -axis", then its gradient is infinite:
this would only happen because of a division by 0 in the original