Question

determine where the function f(x) is continuous

(x^2+7)/(x^2+x-6)

Answer 1

The question here is where is it _not_ continuous?

Answer 2

That's at the points where it's undefined, or makes a jump.

Answer 3

This function has undefined points. Can you see why? It's because the denominator is going to be 0 sometimes.

Answer 4

If that is a rational function (a fraction) then we find where the denominator is equal to zero. Whenever the denominator is equal to 0 we have a discontinuity. That is the function is not defined because we cannot divide by the number 0.

Answer 5

so this is not ever going to have the denominator = zero?

Answer 6

So we solve for x^2+x-6=0 we can use the quadratic equation or factor it.

Answer 7

Exactly. Thus, we are looking for when the the denominator is equal to zero and exclude (do not include) this for the domain (allowed values of x).

Answer 8

Can you solve for x^2+x-6=0?

Answer 9

so basically this is never continuous?

Answer 10

Not quite. This function is continuos everywhere except when x^2+x-6=0 (i.e whichever x values satisfy the equation x^2+x-6=0)

Answer 11

-3,2

Answer 12

exactly!!!

Answer 13

:-) awesome

Answer 14

So we conclude: the function f(x) is continuous everywhere except when x=-3 and x=2

Answer 15

hmm, this concept confuses me! what happens to the numerator? it doesn't matter?

Answer 16

this is a better way to look at it. When does a fraction make sense?
a/b makes sense whenever b is not 0. a can be any number, since we can divide any number by another (except for 0)

Answer 17

where the function is continuous means for which x-values does our function defined (for which x-values can we evaluate our function)...we cannot evaluate the function when we divide by 0 i.e. when the denominator is 0 b/c we don't know how to divide by 0

Answer 18

***is our function defined (not "does our function defined)

Answer 19

exactly the numerator can equal anything because whichever values of x we plug in we can evaluate it

Answer 20

hmm okay! I hope I can burn this concept into my brain!