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

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Introduction

Functions

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

  • y = 2x + 1

image00.png

image09.png

image26.pngimage17.png

image32.png

image37.pngimage35.pngimage36.png

  • image01.pngimage38.pngimage02.pngimage03.png

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 Rimage04.pngimage05.png

image39.png

It is called the domain.

image40.png

f:x --> 2x + 1image41.pngimage42.png

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

image13.pngimage12.pngimage11.pngimage10.pngimage08.pngimage07.pngimage06.png

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)

...read more.

Middle

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)

...read more.

Conclusion

#160;     sinAcosA + sinAcosA

                =        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

dydyduimage31.pngimage30.png

dx        du        dx

When y = [f(x)]n

dy        f'(x) n [f(x)]n-1image30.png

dx        

dy  1  image30.png

dx                dy/dx

The Product Rule

If y = f(x)g(x)

dy             v du        u dvimage30.png

dx               dx           dx

The Quotient Rule

If y =          f(x)

        g(x)

dy             v du        u dvimage30.pngimage33.png

dx               dx           dximage34.png

                    v2

Differentiating ef(x)

If y = ef(x)

dy               f'(x)ef(x)image30.png

dx                

Differentiating ln(x)

If y = ln(x)

dy1image30.png

dx                x

If y = ln(f(x))

dyf'(x)image30.png

dx                f(x)

∫x-1 dx = ln(x)

Differentiating Trig Functions

If y = sinx

dy        cosximage30.png

dx

If y = cosx

dy        -sinx

dx

If y = tanx

dy   1   _                sec2ximage30.pngimage30.png

dx        cos2x

If y = secx

dy sinx _                secxtanximage30.pngimage30.png

dx        cos2x

If y = cossecx

dy cosx                -cosecxcotximage30.pngimage30.png

dx        sin2x

If y = cotx

dy  -1  _                -cosec2ximage30.pngimage30.png

dx        sin2x

...read more.

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