How do you dind the vertical and horizontal asymptotes, domain, range, and roots of f(x)=−1/x−3+2 showing work?

Question

How would you solve this while showing work?

Answer 1

If your given function is f(x)=-1/(x-3)+2 then first you should start by identifying the asymptotes of the function by testing the limits of the variables. Once you find the asymptotes you can determine the domain and range. Finally the roots can be determined by setting f(x)=0 and finding the x value that fulfills this.

1. Identifying the asymptotes:
2. First we should locate where our variables are, in this case we have one variable x in the denominator of our fraction -1/(x-3).
3. Since division by 0 is impossible, we can find our vertical asymptote by finding the value of x that causes x-3=0. Thus x=3 is our vertical asymptote.
4. For the horizontal asymptote we test the limits of our function using -∞,∞, and the vertical asymptote from its left (negative) and right (positive) sides.
5. lim_x→-∞f(x)=-1/(-∞-3)+2=-1/-∞+2=0+2=2. Hence one horizontal asymptote is y=2.
6. lim_x→∞f(x)=-1/(∞-3)+2=-1/∞+2=0+2=2. Which still returns an asymptote of y=2.
7. lim_x→3-f(x)=-1/(-0)+2=∞+2=∞. Hence y goes on to positive infinity. (-0 represents negative numbers getting smaller as they approach 0)
8. lim_x→3+f(x)=-1/(+0)+2=-∞+2=-∞. Hence y goes on to negative infinity. (+0 represents positive numbers getting smaller as they approach 0)
9. Determining the domain and range: Now that our asymptotes and limits are defined we can set the domain and range.
10. For the domain we find what x values are included in the function. Since the only vertical asymptote is x=3 and x cannot be exactly 3 then the domain is (-∞,3)υ(3,∞).
11. For the range we find what y values are included in the function. Since the only horizontal asymptote is y=2; y cannot be exactly 2; and we identified that y can continue to -∞ and ∞ then the range is (-∞,2)υ(2,∞).
12. Determining the roots: Finally the roots can be found by setting f(x)=0.
13. -1/(x-3)+2=0 → -1/(x-3)=-2 → 1=2(x-3) → 1=2x-6 → 7=2x → x=3.5. Therefore our only root is x=3.5

Answer 2

I;m assuming the function you are referring to is -1/(x-3) +2 which is different mathematically from the equation you wrote which would read 1/x -1 if simplified. Parehtnesis/brackets are very important especially when typing out the answer to avoid confusion. Now in orde to find the y intercept, you must set the x =0 and you would get -1/-3 + 2 giving you a solution of y= 7/3 the y intercept is at (0,7/3) in order to find the x intercept the opposite must be true,set y= 0 you would get 0= -1/(x-3) +2 so -2 =-1/(x-3) --> (-2(x-3)) =-1/-2 --> (x-3)=1/2--> x= 1/2 +3 so the x value is 7/2, therefore the x intercept is (7/2,0)