Maths A-Level C3 Notes
Algebraic Fractions
An algebraic fraction can always be expressed in different, yet equivalent forms. A fraction is expressed in its simplest form by cancelling any factors which are common to both the numerator and the denominator.
Algebraic Fractions can be simplified by cancelling down. To do this, numerators and denominators must be fully factorised first. If there are fractions within the numerator/denominator, multiply by a common factor to get rid of these and create an equivalent fraction:
To multiply fractions, simply multiply the numerators and multiply the denominators. If possible, cancel down first. To divide by a fraction, multiply by the reciprocal of the fraction:
To add or subtract fractions, they must have the same denominator. This is done by finding the lowest common multiple of the denominators:
When the numerator has the same or a higher degree than the denominator (it is an improper fraction), you can divide the terms to produce a mixed fraction:
Functions
Functions are special types of mappings such that every element of the domain is mapped to exactly one element in the range. This is illustrated below for the function f (x) = x + 2
The set of all numbers that we can feed into a function is called the domain of the function. The set of all numbers that the function produces is called the range of a function. Often when dealing with simple algebraic function, such as f (x) = x + 2, we take the domain of the function to be the set of real numbers, ℝ. In other words, we can feed in any real number x into the function and it will give us a (real) number out. Sometimes we restrict the domain, for example we may wish to consider the function f(x) = x + 2 in the interval -2 < x < 2.
Consider the function f (x) = x2. What is the range of f (x)? Are there any restrictions on the values that this function can produce? When trying to work out the range of a function it is often useful to consider the
An algebraic fraction can always be expressed in different, yet equivalent forms. A fraction is expressed in its simplest form by cancelling any factors which are common to both the numerator and the denominator.
Algebraic Fractions can be simplified by cancelling down. To do this, numerators and denominators must be fully factorised first. If there are fractions within the numerator/denominator, multiply by a common factor to get rid of these and create an equivalent fraction:
To multiply fractions, simply multiply the numerators and multiply the denominators. If possible, cancel down first. To divide by a fraction, multiply by the reciprocal of the fraction:
To add or subtract fractions, they must have the same denominator. This is done by finding the lowest common multiple of the denominators:
When the numerator has the same or a higher degree than the denominator (it is an improper fraction), you can divide the terms to produce a mixed fraction:
Functions
Functions are special types of mappings such that every element of the domain is mapped to exactly one element in the range. This is illustrated below for the function f (x) = x + 2
The set of all numbers that we can feed into a function is called the domain of the function. The set of all numbers that the function produces is called the range of a function. Often when dealing with simple algebraic function, such as f (x) = x + 2, we take the domain of the function to be the set of real numbers, ℝ. In other words, we can feed in any real number x into the function and it will give us a (real) number out. Sometimes we restrict the domain, for example we may wish to consider the function f(x) = x + 2 in the interval -2 < x < 2.
Consider the function f (x) = x2. What is the range of f (x)? Are there any restrictions on the values that this function can produce? When trying to work out the range of a function it is often useful to consider the