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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
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)
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)
Conclusion
#160; sinAcosA + sinAcosA
...read more.
= 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
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