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

Reading it as f(x) = -1/(x - 3) + 2, this is the basic reciprocal function 1/x that has been flipped, shifted right 3 and shifted up 2. Once you see it that way, every part of the question falls out of the transformations, but let me show the work for each piece anyway.

For the vertical asymptote, ask where the function is undefined. A fraction blows up when its denominator is zero, so set the denominator equal to zero:

x - 3 = 0

x = 3

So the vertical asymptote is x = 3.

For the horizontal asymptote, think about what happens as x gets very large in either direction. The piece -1/(x - 3) gets closer and closer to zero, because you are dividing by something enormous. That leaves just the +2 behind:

y = 2

For the domain, the only thing that can go wrong is dividing by zero, and we already found that happens at x = 3. So the domain is every real number except 3, which in interval notation is (negative infinity, 3) union (3, infinity).

For the range, notice that -1/(x - 3) can be any number except zero, since a fraction with 1 on top never actually reaches zero. Adding 2 to every value shifts that gap up to 2. So the range is every real number except 2, or (negative infinity, 2) union (2, infinity).

Remember, roots are where the function equals zero, so set f(x) = 0 and solve:

-1/(x - 3) + 2 = 0

Subtracting 2 from both sides:

-1/(x - 3) = -2

Multiplying both sides by (x - 3):

-1 = -2(x - 3)

Dividing by -2:

1/2 = x - 3

x = 3.5

So, our final answers are a vertical asymptote at x = 3, a horizontal asymptote at y = 2, a domain of all reals except 3, a range of all reals except 2, and a single root at x = 3.5.

If you notice, the asymptotes are exactly where the graph got shifted to. The 3 inside the parentheses moved the vertical asymptote from x = 0 over to x = 3, and the +2 on the outside lifted the horizontal asymptote from y = 0 up to y = 2. Once you spot that, you can read both asymptotes straight off the equation without any work at all.

Answer 2

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 3

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)