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Algebraic Functions: Domain and Range, Even – Odd functions – Operation/Combination

We have already discussed about algebraic functions earlier but it being a vast topic there are few associated chapters to it which are equally important to be learnt about. We will see those here in details.

Firstly, as we know, there are two ends of a relation and function, domain and range. Determining domain and range or determining whether the relation is a function from a given set of domain and range is also necessary.

For example,

{(3 , 4), (2 , 5), (4, 6), (3, 9)} Determine if this relation is a function or not.

As we know a function is a pair of unique values of x (domain) & y (range). But, also y can be of the same values if it corresponds to different values of x. But in the above problem we find there are two different values of range for same value of domain. Hence this relation is not a function.

Whereas {(3, 5), (2, 5), (-3, 5), (-2, 5)} is a function as there is a different domain for the range.

Next, is to find out if a given function is even or odd.
An even function is a function that is same for both ‘x’ & ‘–x’. And going by the opposite, an odd function is the function which is different for ‘x’ and ‘–x ‘. There are two ways to determine if a function is even or odd. You can do it either graphically or algebraically.
If it’s a symmetrical graph along the x-axis that is the graph intersects the x-axis at same coordinates for both positive and negative values then it’s an even function, otherwise odd.

To solve this algebraically, we need to plug in (–x) in place of (x) and if the result is same as the given function then it is an even function or else odd. For example,

f(x) = –5x2 + 6, so if we put (–x) in place of (x) we get;
f(–x) = –5(–x )2 + 6 = –5x2 + 6     which is same as the given function hence this is an even function.

Finally, we should be well versed with normal mathematical operations i.e. addition, subtraction, multiplication and division with functions also known as combination of functions.

Given f(x) = x + 2 and g(x) = 2 – 3x, find f(x) + g(x), f(x) – g(x), f(x) ∙ g(x) and f(x)/g(x)

f(x) + g(x) = (x + 2) + (2 – 3x) = 4 – 2x
f(x) – g(x) = (x + 2) – (2 – 3x) = 4x
f(x) · g(x) = (x + 2) (2 – 3x) = 4 – 4x – 3x2
f(x) / g(x)  = (x + 2) / (2 – 3x)

So, basically, operation of functions or combining functions is nothing but simple mathematical operation of the given expressions i.e. functions of x.

This would be a comprehensive coverage of algebraic function related topics.