Explain how you can tell(without graphing) that the function has no x intercept and no asymptotes. What is the end behaviour?
r(x)= x^6+10 / x^4+8x^2+15
How would I solve this? I know it has something to do with factoring but im not sure what else to do.
this is what i have so far:
r(x)= x^6+10 / (x^2+3)(x^2+5)
Can someone help me finish? and explain how the answer? thanks
Good start!
We will examine the numerator, x^6+10.
It is a monotonically increasing function, the minimum value of which is 10 when x=0. So it is non-negative over its domain, ℝ.
The factors of the denominator have similar properties, non-negative throughout its domain, ℝ.
Since the function is a non-negative number divided by a non-negative number, there are no vertical asymptotes, nor does it cross the x-axis.
Sorry, there is a correction:
"It is a monotonically increasing function..."
should read
"It is an even function..."
The arguments and conclusions do not change.
To determine if the function has any x-intercepts, we need to see if there are any values of x for which the numerator, x^6 + 10, equals zero.
However, we cannot solve this equation by factoring since the exponent of x is too high. Instead, we can set the numerator equal to zero and try to solve it.
Setting x^6 + 10 = 0, we subtract 10 from both sides:
x^6 = -10.
Since x^6 cannot be negative for any value of x, there are no solutions for x, which means the function has no x-intercepts.
Now let's analyze the denominator, (x^2 + 3)(x^2 + 5), to see if there are any asymptotes. To find the vertical asymptotes, we need to find values of x for which the denominator equals zero.
Setting (x^2 + 3)(x^2 + 5) = 0:
x^2 + 3 = 0 or x^2 + 5 = 0.
Subtracting 3 from both sides, we get:
x^2 = -3, which has no real solutions.
Subtracting 5 from both sides, we get:
x^2 = -5, which also has no real solutions.
Since there are no values of x for which the denominator equals zero, the function does not have any vertical asymptotes.
Finally, to determine the end behavior of the function as x approaches positive or negative infinity, we examine the highest power of x in both the numerator and the denominator.
In this case, the highest power of x in the numerator is x^6, and the highest power of x in the denominator is x^4. Since the degree of the numerator is higher than the degree of the denominator, the function has no horizontal asymptotes.
To summarize:
- The function has no x-intercepts.
- The function has no vertical asymptotes.
- The function has no horizontal asymptotes.
- The end behavior of the function is determined by the higher power of x in the numerator (x^6). As x approaches positive or negative infinity, the function grows without bound.
You have correctly factored the function as r(x) = x^6 + 10 / (x^2 + 3)(x^2 + 5).
To determine whether a function has x-intercepts (also known as zeros), you need to find the values of x for which the function equals zero. Similarly, to determine the presence of asymptotes, you need to analyze the behavior of the function as x approaches positive or negative infinity.
Given the function r(x) = (x^6 + 10)/(x^4 + 8x^2 + 15), you have correctly factored the denominator as (x^2 + 3)(x^2 + 5). Now, let's continue analyzing the function and determining if there are any x-intercepts or asymptotes.
To find x-intercepts, set the numerator equal to zero and solve for x:
x^6 + 10 = 0
Unfortunately, this equation does not have any real solutions. The function, therefore, does not have any x-intercepts.
To determine the presence of asymptotes and analyze the end behavior, we consider the degrees of the numerator and denominator.
The degree of the numerator is 6, which is greater than the degree of the denominator, which is 4. This implies that as x approaches positive or negative infinity, the function r(x) also approaches positive or negative infinity, respectively. In other words, r(x) has no horizontal asymptotes.
To summarize:
- The function r(x) = (x^6 + 10)/(x^4 + 8x^2 + 15) does not have any x-intercepts.
- The function does not have any horizontal asymptotes.
- The end behavior of the function is as x approaches positive or negative infinity, the function also approaches positive or negative infinity, respectively.