Functions

Mappings transform one set of numbers into another set of numbers. We could display a mapping in three ways

• y = 2x + 1

y = x2                 is an example of a many-to-one mapping

y = ±√x        is an example of a one-to-many mapping

Functions only have one output for each input. They are one-to-one or many-to-one mappings.

Function Notation

f(x) = 2x + 1                xЄ R

It is called the domain.

f:x --> 2x + 1

The set of outputs is called the range.

f(x) = x2        xЄ R

is f(x) ≥ 0        or         f ≥ 0

BUT NOT x ≥ 0

We can think of the domain as the values on the x axis and the range as the values on the y axis.

It is often easiest to find the range by sketching the graph.

Sometimes we need to limit a domain so that all the inputs give an output (ie can't square root a negative number and cannot divide by zero.

f(n) means find the value of f(x) when x = n

Check that roots of the equation lie within the domain, otherwise reject them

Composite Functions

f(g(x)) we write as fg(x)

Do g(x) first then f(x) !!!

Note that in general fg(x) ≠ gf(x)

Inverse Functions

If a function is one-to-one then an inverse function exists, called f-1(x)

An asymptote is formed where there is a "forbidden" number. For the domain, a vertical asymptote is formed. For the range, a horizontal asymptote is formed.

To find the inverse of a quadratic function, swap x and y and complete the square.

Algebraic Fractions

Simplifying

To simplify

• Look for common factor in the numerator and denominator
• Sometimes we have to factorise first
• If we have fractions in the numerator or denominator, multiply to remove the fraction

Multiplying and Dividing

• Multiply the numerators and multiply the denominators.
• Cancel where necessary
• Where one fraction is divided by another, invert/flip the second fraction and multiply.

Adding and Subtracting

• To add or subtract fractions, they must have the same denominator
• Be careful when subtracting (especially where there are double negatives)
• When the denominators have no common factor, their product gives the new denominator (multiply the numerator accordingly)
• It may be necessary to factorise the denominators first to spot the common factors

Writing Improper Fractions as Mixed Numbers

• Improper fractions, where the numerator is larger than the denominator can be written as mixed numbers
• The same can be done with algebraic fractions, where the numerator can order that is equal to or larger than the denominator

• The remainder is written as a fraction

• There are two methods for doing this

• Polynomial long division
• Remainder theorem

• Remainder has to have a lower power than the divisor

        Let F(x)

                =        2sinAcosA

Cos(2A)        =        cosAcosA - sinAsinA

                =        2cos2A - 1

                =        1 - 2sin2A

Tan(2A)        =         tanA + tanA

                        1 - tanAtanA

                =           2tanA  

                        1 - tan2A

Can calculate sin(3A) by expanding sin(2A + A)

((sin(3A) = 3sinA – 4sin3A))

Expressing aSinθ +  bCosθ as a Single Sin or Cos

aSinθ +  bCosθ Ξ Rsin(θ + c)

Rsinc = b

Rcosc = a

R = √a2 + b2

R is the min/max value (with -R being the max/min value)

Differentiation

We call (function)n a function of a function.

Chain Rule

dydydu

dx        du        dx

When y = [f(x)]n

dy        f'(x) n [f(x)]n-1

dx        

dy  1  

dx                dy/dx

The Product Rule

If y = f(x)g(x)

dy             v du        u dv

dx               dx           dx

The Quotient Rule

If y =          f(x)

        g(x)

dy             v du        u dv

dx               dx           dx

                    v2

Differentiating ef(x)

If y = ef(x)

dy               f'(x)ef(x)

dx                

Differentiating ln(x)

If y = ln(x)

dy1

dx                x

If y = ln(f(x))

dyf'(x)

dx                f(x)

∫x-1 dx = ln(x)

Differentiating Trig Functions

If y = sinx

dy        cosx

dx

If y = cosx

dy        -sinx

dx

If y = tanx

dy   1   _                sec2x

dx        cos2x

If y = secx

dy sinx _                secxtanx

dx        cos2x

If y = cossecx

dy cosx                -cosecxcotx

dx        sin2x

If y = cotx

dy  -1  _                -cosec2x

dx        sin2x