Question

Describe the behavior of the graph at the x-intercept for the function f(x)=(2x-7)^7(x+3)^4
I know what to do I just need a little guidance and have a couple of questions.
Please help I will fan and medal!!!

Answer 1

Im not sure if this is a question where you will list the doman, range, x intercept and all of that other stuff

Answer 2

or if you are just listing the x intercepts

Answer 3

I think they want to know if the graph is rising, or decreasing, etc. at the x-intercepts

Answer 4

Oh okay, but the graph is so weird

Answer 5

Could you help me with finding the domain and range?

Answer 6

@phi ^^^^

Answer 7

I can do everything else

Answer 8

f(x)=(2x-7)^7(x+3)^4

domain: you start by assuming the domain is all x
then remove any "divide by zero" or "root of a negative number"
we don't have any divides, and no square roots. so we can keep all x values for the domain

Answer 9

Okay, i get that
Isn't the domain for this specific thing suppose to be something in the form of (0,infinity) (that was just a random example)

Answer 10

I was thinking that the domain is, (0, positive infinity)

Answer 11

f(x)=(2x-7)^7(x+3)^4

range: if x is "very negative"
then 2x -7 will be negative. Raising it to an odd power (7 in this case), it stays negative.
In other words, as we make x more negative, f(x) gets more negative. f(x) can move as far as we want toward negative infinity.

on the other hand, if x is "big positive"
2x-7 is positive
raising to 7th power , it's still positive

ditto for x+3 and (x+3)^4
i.e. we can make f(x) move toward + infinity

thus the range is
\[ (-\infty , +\infty) \]

Answer 12

the domain is all x

\[ (0, +\infty) \]
leaves out negative x values, so that is not the entire domain.

Answer 13

Ohhhh okay this makes sense

Answer 14

@phi so for range would it be (-infinity, positive infinity)?

Answer 15

And I now understand, if it were (0, infinity) it would exclude the -3 point that it lays on and that wouldn't make sense

Answer 16

yes, both the domain and range are -infinity to + infinity

Answer 17

wolfram can graph this function
http://www.wolframalpha.com/input/?i=f(x)%3D(2x-7)%5E7(x%2B3)%5E4

Answer 18

@phi Okay, thank you so much!!!

Answer 19

I was using desmos lol

Answer 20

@phi sorry I have two more questions
For the increase would you say (0, +infinity), and for the decrease you would say (-3, 0)?

Answer 21

The curve f(x) rises up to the x-axis at x=-3 then turns back down. the x-axis is tangent to the curve at x= -3

at the other x-intercept, the curve rises rapidly until it is just below the x-axis at about x=2, then rises very slowly, crossing the x-axis at x= 3.5. At about x= 4.5, the curve's slope rapidly increases, and it rises rapidly

Answer 22

@phi okay so the increase would be (-3,0),(0,4.5)?

Answer 23

the curve increases from -infinity up to -3
and then from 3.5 to + infinity

in between -3 and 3.5, the curve dips down and back up

Answer 24

@phi Oh okay, I get it now