Question

Which of the following functions is not continuous for all real numbers x ?
A) f(x) = x^1/3
B) f(x) = 2/(x + 1)^4
C) f(x) = |x + 1|
D) f(x) = √(1 + e^x)
E) f(x) = x − 3/(x^2 + 9)
How do i figure out if a function is continuous at all real numbers?

Answer 1

if a function is continuous it is defined for every value of x
we can also say that we can draw the graph of a continuous function without lifting our pen for example -this function is continuous

Answer 2

what method would i use to find that is has continuity

Answer 3

are you allowed to use graphing calculators?

Answer 4

No

Answer 5

If you see that a function is undefined at a particular x-vaue, then the function is not continous.

Answer 6

For example,

\(\large\color{black}{ \displaystyle y=\frac{1}{x-3} }\)

is not continous, because the function is undefined at x=3.
(you end up dividing by 0)

Answer 7

how did you get that?

Answer 8

That is an example, not an answer...

Answer 9

I know

Answer 10

Can you divide by 0?

Answer 11

No...

Answer 12

Yes, you can NOT!

good!

When, the function: \(\large\color{black}{ \displaystyle y=\frac{1}{x-3} }\)

is evaluated at x=3, you get: \(\large\color{black}{ \displaystyle y=\frac{1}{\color{red}{3}-3} =\frac{1}{0}\color{blue}{=\rm UNDEFINED} }\)

Answer 13

So if you plug zero into the equation and it's undefined then it not continuous?

Answer 14

So when x=3, your output (y) is undefined.
Therefore, the function is not continous because it has a hole at x=3.

Answer 15

Wel, if for some vaues of x, let's say x=c, you get an undefined output,
THEN this means that the function is not continous because it has a gap at x=c.

Answer 16

But I'm not given a point...

Answer 17

you know that in a case of, (lets make a new function up)

y=1/(x+5)

Wha happenes when you plug in x=-5 into this function?

Answer 18

you get undefined

Answer 19

yes, exacty that means that you can't have a point for x=-5,
right?

Answer 20

in other words, there is nothing there on the vertical of x=-5.

Answer 21

And that means you have a gap for x=-5.
And this means the function is not continous (because of the hole at x=-5)

Answer 22

but i just trying to figure out which ONE is not continuous

Answer 23

Ok, analyze each function in the answer-choices.

You might have an "undefined" output, for a coupe of reasons.

1) Negative in the square root (negative in the 4th roots, or in any even root)
2) When you end up dividing by 0

Answer 24

Note that negative inside the cube, fifth or any ODD rood, would still be defined

Answer 25

lets do option 1 together

Answer 26

A) f(x) = x^1/3
B) f(x) = 2/(x + 1)^4
C) f(x) = |x + 1|
D) f(x) = √(1 + e^x)
E) f(x) = x − 3/(x^2 + 9)

I wil post a options for reference

Answer 27

\(\large\color{#000000 }{ \displaystyle x^{1/3}}\)

Answer 28

Will you end up "dividng by zero" if this function does not even have a denominator??

Is this function even undefined if you pug lets say -8 or -1 for x, or if you plug anything else into this function?

Answer 29

no

Answer 30

No to which question(s), the first, the second or both?

Answer 31

both

Answer 32

very good, so that means that this function is _____ ?

Answer 33

the function is continuous

Answer 34

Yes, very good, answer choice A is continous

Answer 35

Which letter do you want to do next?

Answer 36

c

Answer 37

\(\large\color{#000000 }{ \displaystyle f(x)=\left|x+1\right| }\)

Answer 38

We, that is just an absolute value
(bars denote an absolute value)

Answer 39

Do you think there is a number that you can plug into the function for x, to make the function undefined ?