Describe the graphs of the functions f(x) = 2^x – 3 and g(x) = –2^x – 3. Compare and contrast the domain and range of f(x) and g(x) Describe the graphs of the functions f(x) = 2^x – 3 and g(x) = –2^x – 3. Compare and contrast the domain and range of f(x) and g(x) @Mathematics

Question

turingtest · Answer

f(x) = 2^x – 3
y-intercept=-2
horizontal asymptote: y=-3
Domain all reals, Range y ≥-3

turingtest · Answer

g(x) = –2^x – 3
y-int=-2
horizontal asymptote=-3
Domain ALL REALS
Range y≤-3

turingtest · Answer

f(x) increases as x increases
g(x) decreases as x increases

anonymous · Answer

Thank you :) how did you find the range?

turingtest · Answer

Imagine if $$x  \to -\infty$$then $$2^x \to 0$$which leave the equations as
f(x)=-3 and g(x)=-3. so they cannot go past -3.

turingtest · Answer

actually I made a slight mistake
since x cannot equal infinity, f(x) and g(x) cannot EQUAL 3, they can only approach it asymptotically, so the ranges should be
f(x): y>-3 and g(x) y<-3

anonymous · Answer

Oh ok, so 3 is like the discontinuity?

turingtest · Answer

Well, a discontinuity is a point where the graph has a "hole" or a "jump". This is not the case here, since the function is continuous. The point is that making x more and more negative only APPROACHES 3, but never equals it.
I presume you have not studied limits yet, but the idea will become more clear when you do.

anonymous · Answer

Mhm, thanks for your help, and sorry im not to good with ranges in graphs haha.

anonymous · Answer

So if the line goes on the number and goes in a certain direction, that is how you find the range?

turingtest · Answer

Yeah many ranges can only be determined through Calculus, but some things to watch for are square roots and exponential functions, like
$$y=\sqrt x$$ has a range of $$[0,\infty)$$because if x<0 y is imaginary (In this case the domain is the same as the range. Think about why.)
Or$$y=a^x$$has a range of $$(0,\infty]$$notice that 0 is not included in the range because in order to get $$y=0$$ we must have$$ x=-\infty$$ which we cannot do (plugging in infinity as a value is generally a no-no in basic mathematics)

turingtest · Answer

I have no idea what you meant by your last question...

turingtest · Answer

The range is the set of all possible values a function can take on. I'm not sure I can describe it much beyond that.

anonymous · Answer

Kk well thank you for your help :)

anonymous · Answer

Oh I totally get it now, I had to graph it -__- lol thank you so much