Question

could somebody explain this part for me, please?

Answer 1

@pencile: could you please be more specific about what you need to know?

Answer 2

I do not understand why some of the functions are acceptable while others are not?
and also, how can I know whether they are single valued, continues, or finite

Answer 3

well, you'd know if they're single valued or continuous or finite, by graphing them :)

acceptable, in this context means, "does the graph of that function extends over the given interval"

Answer 4

so , I have to replace both two intervals with X

Answer 5

nope, the interval is the range of the function, so acceptable is, "is that a valid range for it"

Answer 6

also, can you show me what the first derivative for one of them please

Answer 7

well, \(\bf \cfrac{d}{dx}[e^{-x}]\implies e^{-x}\)

Answer 8

@jdoe0001 Woops! Chain rule, ya? :o

Answer 9

hmmm is it meant to be? was thinking about it, but dunn of "x" is meant to be differentiable or just a variable

Answer 10

For the last one (e), it says not acceptable because the first derivative is not continuous. what does it mean, can you show me please

Answer 11

hmm mm @pencile dunno that one myself, well, not the absolute value that is
I'm thinking zepdrix may though

Answer 12

thank you very much, i really appreciate your help

Answer 13

Recall how we define absolute value:

When x is greater than 0,
our absolute value x is giving us x,

When x is less than 0,
our absolute value x gives us -x,\[\Large e^{-\color{orangered}{|x|}}=\cases{e^{-\color{orangered}{x}},\qquad\qquad x\ge0\\ e^{-\color{orangered}{(-x)}},\qquad\quad x<0}\]

So on the right side of of the y-axis we have the function e^-x