• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

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

Extracts from this document...

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.

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related AS and A Level Core & Pure Mathematics essays

  1. Marked by a teacher

    The Gradient Function

    5 star(s)

    6x1 6x1 y=3x3 3(3x3-1) 9x2 9x2 y=3x4 3(4x4-1) 12x3 12x3 "Odd functions" Curves of y= axn Gradient Function x1/2 0.5x-0.5 x1/4 0.25x-0.75 x-1 -x-2 x-2 -2x-3 -2x2 -4x It is clear to see that the overall equation for the gradient function of this, or any curve y=axn is naxn-1.

  2. The open box problem

    To check this I will now draw a graph, which will hopefully show in more detail the maximum volume and value for x. The graph shows that the maximum volume is about 41.64 and x is 1.264 ( at the bottom it says the co-ordinates of where the cursor was hovered).

  1. Numerical integration can be described as set of algorithms for calculating the numerical value ...

    If the number of rectangles is doubled M2n the height of each is halved (h/2). According to www.enm.bris.ac.uk the absolute error is proportional to h2. Absolute error ? h2 Absolute error = kh2 As a result in the first situation where Mn rectangles are used each of height h, the error is Mn= kh2.

  2. MEI numerical Methods

    I will use one root a as 0 and the other root b as ?/4. Bearing this in mind we can substitute the values into the formula: The formula is the same as the secant method however remember the false position method roots approximation must support the sign change argument.

  1. Math Portfolio Type II - Applications of Sinusoidal Functions

    = 0.846 sin[0.015(n -127.798)] + 6.362. This is found through using the TI-84 Plus graphing calculator. All the coordinates are listed in a table and the equation is found using sinusoidal regression, which is done with the graphing calculator. 6.

  2. Numerical solutions of equations

    The error bound is: 0.0000000005 However, the Newton-Raphson method would not work with the graph below: If I start from the turning point of the curve, I do not think it will converge (see Figure 4). The equation for this function is 0 = x4+2x3+3x-4 f'(x)

  1. Math assignment - Families of Functions.

    The y-intercept for an absolute value function is b, and it is not to be forgotten that b is inside the absolute value symbols. So, you calculate the graph through using the three points that you have got, the y-intercept, the vertex, and the point, symmetric to the y-intercept.

  2. C3 COURSEWORK - comparing methods of solving functions

    method to find the same roots. y= x³+3x²–3 By using Decimal search: Find the root in the interval [0, 1], n x f(x)=x³+3x²–3 1 0 -3 2 0.1 -2.969 3 0.2 -2.872 4 0.3 -2.703 5 0.4 -2.456 6 0.5 -2.125 7 0.6 -1.704 The root lies between 0.8 and

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work