Prove: e is an irrational number. the following is an outline of a proof: (a) use the taylor formula to show 0 < e − 1 + 1 1! + 1 2! + 1 3! + + 1 n! < 3 (n + (b) assume to the contrary that e is rational, say, e = a b , where a and b are positive integers. pick n > b and n ≥ 3. substitute e = a b into the inequality in (a) and multiply the inequality by

the required function is .

step-by-step explanation:

from the given table it is noticed that the graph passing through (-3,7), (0,4), (3,1) and (8,-4).

it is a linear function because the value of y is decrease by one as the value of x increase by 1.

the slope of the function is

the slope of the function is -1 and the y-intercept is 4.

the slope intercept form is

where, m is slope and b is y-intercept.

therefore the required function is . the graph of this function is shown below.

d

step-by-step explanation:

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step-by-step explanation: