Let X = { 1, 2, 3, 4 }. Examine whether each of the relations given below is a function from X to X or not. Explain.
(i) f = { (2, 3), (1, 4), (2, 1), (3, 2), (4, 4) }
(ii) g = { (3, 1), (4, 2), (2, 1) }
(iii) h = { (2, 1), (3, 4), (1, 4), (4, 3) }

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Solution

$(i)$ The element $2$ have two images therefore $f$ is not a function.
$(i i)$ Every element of $X$ does not have an image therefore $g$ is not a function.
$(i i i)$ Every element of $X$ has a unique image therefore $h$ is a function.

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I f I_{1} = \int_{0}^{1} 2^{x^{2}} d x, I_{2} = \int_{0}^{1} 2^{x^{3}} d x, I_{3} = \int_{1}^{2} 2^{x^{2}} d x a n d I_{4} = \int_{1}^{2} 2 x^{3} d x t h e n

\int_{- 1}^{4} f (x) d x = 4

\int_{2}^{4} [3 - f (x)] d x = 7 t h e n \int_{- 1}^{2} f (x) d x =

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(ii)

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