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 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 graph of the function, this is shown left. We can see that the function only gives out positive numbers (x2 is always positive for any real number x). There are no further restrictions. We can see that f can take any positive value, therefore the range of f is the set of all positive numbers, we may write f(x) ≥ 0.
When each of the elements of the domain is mapped to a unique element of the range, under a mapping, the mapping is said to be one-to-one. When two or more elements of the domain are mapped to the same element of the range under a mapping, the mapping is said to be many-to-one. Below are two examples. The mapping f is one-to-one, the mapping g is many-to-one.
We need to define more precisely what we mean by a ‘function’.
We can define a function as a rule that uniquely associates each and every member of one set with a member or members of another set. This means that every element of the domain is mapped to an element of the range such that the image of any element in the domain is unique. In other words, each and every element of the domain must be mapped to one and only one element of the range. For example, consider the expression .
Notice that any value of x in the domain, except x = 0 , (i.e. any positive real number) is mapped to two different values in the range. Therefore is not a function.
When looking for the domain of a function, look out for values that would leave a negative root or 0 on the bottom of a fraction. At these values of x, y is undefined.
not a function
Many mappings can be made into functions by changing the domain. For example, the ‘root of x’ mapping can be made into a function by changing the domain from all real numbers, to all positive numbers. This will cut off the bottom half of the graph, meaning every element in the domain is matched uniquely to an element in the range.
Consider the function, g (x) = (x - 2)2. If we...
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