Find the x- and y-intercepts of the rational function. (If an answer does not ex

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Find the x- and y-intercepts of the rational function. (If an answer does not exist, enter DNE.)

t(x) = 
x2 − 2x − 24
x − 7
x-intercept    
(xy) = 
 
 
     (smaller x-value)
x-intercept    
(xy) = 
 
 
     (larger x-value)
y-intercept    
(xy) = 
 
 
Jul 15th, 2015

Thank you for the opportunity to help you with your question!

In order to find the x-intercepts we make t(x) zero and we solve for x like this.

t(x) = (x^2 - 2x - 24)/(x-7) -----------> 0 = (x^2 - 2x - 24)/(x-7) ---------> 0*(x-7) = (x-7)*(x^2 - 2x - 24)/(x-7)

0 = x^2 - 2x - 24  ------------> x^2 - 2x - 24 = 0. This is a quadratic equation that we can solve by factoring. For that, we find two numbers that their product is - 24 and their algebraic addition give - 2. Then the numbers would be:
-6 and 4 since (-6)(4) = - 24 and  - 6 + 4 = -2. Then it would factor like this:

x^2 - 2x - 24 = 0 ---------> (x - 6)(x + 4) = 0. So we equal each factor to zero and solve for x.

x - 6 = 0 ----------> x - 6 + 6 = 0 + 6  ------------> x = 6

x + 4 = 0 ----------> x + 4 - 4 = 0 - 4  ------------> x = - 4

Then we write the oredered pairs (x , y) which are: (6 , 0)  and (- 4, 0)

On the other hand, in order to find the y-intercept we make the x zero like this.

t(x) = (x^2 - 2x - 24)/(x-7)   ----------->  t(0) = (0^2 - 2(0) - 24)/(0-7) -------------> t(0) = (0 - 0 - 24)/(0 -7)

t(0) = -24/-7  ---------> t(0) = 24/7. Finally, we write the ordered pair (x ,y) which is:  (0 , 24/7)

Please let me know if you have any doubt or question :)

Please let me know if you need any clarification. I'm always happy to answer your questions.
Jul 15th, 2015

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