Question

9. A relation contains the points ( -5,-10),( -2,-4)(-1,-2),(4,8) and (5,10) . Is this a function? Explain. (2 points)

Answer 1

Yes it is a function given by f(x) = 2x

Explanation:

Any function can approximated as series or a polynomial. For example,

`e^(x) = 1 + (x)/(1!) + (x^(2))/(2!) + (x^(3))/(3!) ...` (exponential function)

(n! or n factorial is equal to n(n-1)(n-2)...3.2.1 ; 3! = 3.2.1 = 6)

and for the series to converge(the sum does not go to infinity), higher order terms must tend to zero.

General form of a polynomial/series:
`f(x) = a + bx +cx^(2) + ...`

For the given set of points, we can start with the straight line equation:

`f(x) = y = a + bx` ........(1)

Let us take two points from the given relation: (-5, -10), (-1, -2)

and put the respective x and y values in equation (1), we get two equations, which we can then solve simultaneously to get values of
`a` and
`b`:

`-10=a-5b` ........(2)

`-2=a-b` ..........(3)

Now (3) - (2) gives us:
`b=2` and putting the value of
`b` in any of the above equation gives us
`a=0`

Hence, we get the equation,
`f(x)=y=2x`

It can be seen that all the given points satisfies this relation and since we get a unique
`y` for every
`x`, we can call this a function.