Graph (x) = -3x+1/x-2 Identify any intercepts, asymptotes, domain and range.
I am having a hard time understanding this. I have taken notes and asked questions, but when I come home to work on it I cannot figure it out. Please Help and Thankyou!
I am having a hard time understanding this. I have taken notes and asked questions, but when I come home to work on it I cannot figure it out. Please Help and Thankyou!
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First is this equation (-3x + 1) / (x - 2) or is it - 3x + 1/ (x - 2)?
It makes a difference. I am going to assume it is the second version.
Domain means what values over all the Reals work in that equation. The problem arises with the fraction. Denominators are not allowed to equal zero, ever. Since x =2 makes the denominator = 0, that value is not allowed in the domain. There are several ways to write that out, usually x with a equals sign and a slash through the sign meaning "does not equal" 2, is sufficient.
Because x cannot equal 2, the vertical line x = 2 is an asymptote.
Range means what value will the function output as you put in all the various possible x values. This is a thinking problem, usually.
If x is a very large negative number, approaching neg infinity, what happens with the function? the fractional part will be negative, and going to zero because of the large value in the denominator. So -3x becomes dominant. If x is a large negative, -3x is a large positive (pos. infinity).
Similar thinking will lead you understand that as x approaches pos. infinity, the function goes to neg. infinity
Now we have to think about what happens as x approaches 2, from both the left (negative) and right (positive) side.
As x approaches 2 from the left side , x is going from + 1 to + 1. The fraction is negative, and large because the denominator is going small. -3x is positive, but not large. So from the left side, the function goes to neg infinity as x approaches 2.
As x approaches 2 from the right, it is going from 3 to 2. The -3x is neg but not large. The fraction is positive, but large. So, as x is close to 2, on the positive sign, the function is near pos infinity
so the range is all reals.
The only thing left are the intercepts.
The y intercept is when x = 0. The value of the function is - 1/2
To find the x intercept. set the function equal to 0 and solve.
-3x + 1/(x-2) = 0
1/(x -2) = 3x (multiply both sides v by the denominator)
1 = 3x(x-2) = 3x^2 - 6x
3x^2 - 6x - 1 = 0
so now use the quadratic equation so solve for x, and using a calculator the x intercepts are -.15 2.15
It makes a difference. I am going to assume it is the second version.
Domain means what values over all the Reals work in that equation. The problem arises with the fraction. Denominators are not allowed to equal zero, ever. Since x =2 makes the denominator = 0, that value is not allowed in the domain. There are several ways to write that out, usually x with a equals sign and a slash through the sign meaning "does not equal" 2, is sufficient.
Because x cannot equal 2, the vertical line x = 2 is an asymptote.
Range means what value will the function output as you put in all the various possible x values. This is a thinking problem, usually.
If x is a very large negative number, approaching neg infinity, what happens with the function? the fractional part will be negative, and going to zero because of the large value in the denominator. So -3x becomes dominant. If x is a large negative, -3x is a large positive (pos. infinity).
Similar thinking will lead you understand that as x approaches pos. infinity, the function goes to neg. infinity
Now we have to think about what happens as x approaches 2, from both the left (negative) and right (positive) side.
As x approaches 2 from the left side , x is going from + 1 to + 1. The fraction is negative, and large because the denominator is going small. -3x is positive, but not large. So from the left side, the function goes to neg infinity as x approaches 2.
As x approaches 2 from the right, it is going from 3 to 2. The -3x is neg but not large. The fraction is positive, but large. So, as x is close to 2, on the positive sign, the function is near pos infinity
so the range is all reals.
The only thing left are the intercepts.
The y intercept is when x = 0. The value of the function is - 1/2
To find the x intercept. set the function equal to 0 and solve.
-3x + 1/(x-2) = 0
1/(x -2) = 3x (multiply both sides v by the denominator)
1 = 3x(x-2) = 3x^2 - 6x
3x^2 - 6x - 1 = 0
so now use the quadratic equation so solve for x, and using a calculator the x intercepts are -.15 2.15
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Asker
You are correct about the domain.
(- infinity, 2) union (2, infinity)
E-G
You are correct about the domain.
(- infinity, 2) union (2, infinity)
E-G
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f(x) = (-3x + 1)/(x - 2)
The domain is {x: x ≠ 2} and the range is {real numbers}.
There is a vertical asymptote of x = 2.
There is an x-intercept of (1/3, 0) and a y-intercept of (0, -1/2)
The domain is {x: x ≠ 2} and the range is {real numbers}.
There is a vertical asymptote of x = 2.
There is an x-intercept of (1/3, 0) and a y-intercept of (0, -1/2)