State the Vertical Asymptote, Horizontal Asymptote, domain, range, x intercept and y intercept for the following Function.
y= (5-x)/(x+3)
I already know that the Vertical asymptote is -3, so that means the domain is everything except -3, but how do I find the Horizontal asymptote, the two intercepts and the range. I appreciate the help
The x intercept is any x for which y=0. The y intercept is any y for which x=0.
x intercept= 5
y intercept= 5/3 or 1 2/3 if you prefer.
For the range, is there anything that y cannot be no matter what x is?
Google has some great results on the first page for a search of horizontal asymptote.
Well, isn't this just a math circus, juggling with asymptotes and intercepts! Let's dive right in:
Vertical Asymptote: You're absolutely right, the vertical asymptote is x = -3. This means that our function gets a bit wobbly as x approaches -3, but it never quite reaches that value.
Domain: Correct again! The domain is indeed all real numbers except x = -3. In other words, your function can take any x values except -3.
Horizontal Asymptote: To find the horizontal asymptote, we examine what happens as x goes to infinity or negative infinity. In this case, the degree of the numerator and denominator is the same (both are 1). Since the degrees are equal, we look at the ratio of the leading coefficients. Here we have -1/1 or simply -1. So, the horizontal asymptote is y = -1.
Range: To determine the range, we consider what possible y values the function can take. Since the denominator x + 3 can never be 0 (except at the vertical asymptote), the only restriction comes from the numerator 5 - x. As x varies from negative infinity to positive infinity, the numerator can take any value from negative infinity to positive infinity. Therefore, the range of the function is all real numbers.
x-intercept: To find the x-intercept, set y = 0 and solve for x. In this case, setting (5 - x)/(x + 3) = 0 will give us x = 5.
y-intercept: To find the y-intercept, set x = 0 and solve for y. Plugging in x = 0 into the equation, we get y = (5 - 0)/(0 + 3) = 5/3.
So, to summarize:
- Vertical Asymptote: x = -3
- Horizontal Asymptote: y = -1
- Domain: All real numbers except x = -3
- Range: All real numbers
- x-intercept: x = 5
- y-intercept: y = 5/3
Hope this clownaround explanation puts a smile on your face and helps you understand the function better!
To find the horizontal asymptote, you need to compare the degrees of the numerator and denominator.
In the given function, the degree of the numerator is 1 (x has a power of 1), and the degree of the denominator is also 1 (x has a power of 1).
When the degrees of the numerator and denominator are the same, the horizontal asymptote can be found by taking the ratio of the leading coefficients of the numerator and denominator.
In this case, the leading coefficient of the numerator is -1 (coefficient of x), and the leading coefficient of the denominator is 1 (coefficient of x).
Therefore, the horizontal asymptote is y = -1/1 or simply y = -1.
For the x-intercept, you need to find the value(s) of x when y = 0.
Setting y = 0 in the equation and solving for x:
0 = (5 - x) / (x + 3)
Multiplying both sides by (x + 3):
0 = 5 - x
Moving x to one side:
x = 5
So, the x-intercept is x = 5.
For the y-intercept, you need to find the value of y when x = 0.
Setting x = 0 in the equation:
y = (5 - 0) / (0 + 3)
y = 5/3
Therefore, the y-intercept is y = 5/3.
Finally, for the range, we consider the limiting behavior of the function as x approaches positive or negative infinity.
As x approaches positive or negative infinity, the value of y approaches the horizontal asymptote, which is y = -1.
Therefore, the range of the function is all real numbers except y = -1.
To summarize:
Vertical asymptote: x = -3
Horizontal asymptote: y = -1
Domain: all real numbers except x = -3
Range: all real numbers except y = -1
x-intercept: x = 5
y-intercept: y = 5/3
To find the horizontal asymptote, you need to compare the degrees of the numerator and denominator. In this case, the degree of the numerator is 1 and the degree of the denominator is also 1. When the degrees are the same, the horizontal asymptote is found by dividing the leading coefficients of the numerator and denominator.
So, for the given function y = (5 - x) / (x + 3), the leading coefficients are 1 for both the numerator and denominator. Therefore, the horizontal asymptote for this function is y = 1.
To find the x-intercept, you set y equal to zero and solve for x. In this case, you have:
0 = (5 - x) / (x + 3)
Since the numerator cannot be equal to zero, solve for the non-zero value of x:
5 - x = 0
x = 5
So, the x-intercept is (5, 0).
To find the y-intercept, you set x equal to zero and solve for y. In this case, you have:
y = (5 - 0) / (0 + 3)
y = 5/3
So, the y-intercept is (0, 5/3).
For the range, you can observe that as x approaches negative infinity or positive infinity, the function approaches the horizontal asymptote, which is y = 1. Therefore, the range of this function is all real numbers except 1.
To summarize:
Vertical asymptote: x = -3
Horizontal asymptote: y = 1
Domain: All real numbers except -3
Range: All real numbers except 1
x-intercept: (5, 0)
y-intercept: (0, 5/3)