Question

Given: f(x)= 7x^3 + 2
A. identify the function as even, odd, or neither.

B. Describe the end behavior of the function.

Answer 1

Find f(-x)

Answer 2

\[f(x)=7x^3+2 \\ f(-x)=7(-x)^3+2\]
what is the cube of (-1)?

Answer 3

1?

Answer 4

\[(-x)^3=(-1x)^3=(-1)^3(x)^3=(-1)^3x^3=(-1 \cdot -1 \cdot -1)x^3\]

Answer 5

-1 times -1 is 1

Answer 6

but -1 times -1 times -1 is -1

Answer 7

ops i squared it on accident

Answer 8

\[(-x)^3=-1x^3 \text{ or just } -x^3\]

Answer 9

but anyways this means \[f(-x)=-7x^3+2\]

Answer 10

But this isn't the same as 7x^3+2 or the opposite right?

Answer 11

Yes

Answer 12

if it was the same as the function we started with would be even
if it was the opposite of the function we started with it would be odd
so yes you are right it is neither

Answer 13

There is a trick to doing polynomials quicker if you want to know

Answer 14

sure

Answer 15

If all the powers in the polynomial are odd, then the function is odd.
If all the powers in the polynomial are even, then the function is even.
If you have a mixture of evens and odds powers, then the function is neither odd or even.
Now a lot of people get confused when they don't see a variable next to one of the numbers. But you can think of your function as f(x)=7x^3+2x^0

Answer 16

3 is odd
0 is even
so f is neither since you have a mixture of even and odd powers

Answer 17

What I just said works for any polynomial...
f(x)=3 is even because again f(x)=3x^0 and 0 is even
f(x)=x is odd because f(x)=x^1 and 1 is odd
f(x)=x^2+3 is even because f(x)=x^2+3x^0 and 2,0 are even

Answer 18

interesting

Answer 19

Do you know what I mean by a polynomial?

Answer 20

yes

Answer 21

\[P(x)=a_0x^0+a_1x^1+a_2x^2+a_3x^3+a_4x^4+a_5x^5+ \cdots a_nx^{n}\]
where n is integer

Answer 22

\[P_1(x)=a_0x^0+a_2x^2+a_4x^4+a_6x^6 \text{ is even } \\ P_2(x)=a_1x^1+a_3x^3+a_5x^5 \text{ is odd }\]

Answer 23

The function I defined as P_1 is even because 0,2,4,6 are even
The function I defined as P_2 is odd because 1,3,5 are odd

Answer 24

I think i got it

Answer 25

lol ok

Answer 26

So knowing that it is odd that should tell you about the opposite ends of our function.

Answer 27

Oops sorry I was thinking about a different function.

Answer 28

We are looking at f(x)=7x^3+2

Answer 29

and that function had neither symmetry

Answer 30

so what does that mean for the end?

Answer 31

What are you allowed to do?
I can do limits with you or you could graph in your calculator.

Answer 32

Are you familiar with the graph of y = x^3 ?

Answer 33

The ends means like what is happening as we go view the graph for really large values of x or really small values of x

Answer 34

oh okay wouldnt it be kind of small values?