Using the function f(x)=4 - 1/(x+6)^2
A.what is the doamin?
b.the range
c. the asymptotes
domain: all reals except -6
range: all real y < 4
asymp: y=4, x = -6
I was thinking that the asymptoes was y4,x= -6, and the domain was all real but the range I was incorrect. Thanks!
To determine the domain of the function, we need to consider any restrictions on the values of x. In this case, the only restriction occurs when the denominator is equal to zero (since division by zero is undefined). So, we need to solve the equation (x + 6)^2 = 0.
(x + 6)^2 = 0
Taking the square root of both sides:
x + 6 = 0
x = -6
Therefore, the domain of the function f(x) = 4 - 1/(x + 6)^2 is all real numbers except x = -6. In interval notation, the domain is (-∞, -6) U (-6, ∞).
To determine the range of the function, we need to look at the behavior of the function as x approaches the extreme values of the domain. In this case, since the denominator (x + 6)^2 is squared, it is always positive. So, as x approaches negative infinity or positive infinity, the fraction 1/(x + 6)^2 approaches zero. Therefore, the range of the function is all real numbers except when the fraction 1/(x + 6)^2 is equal to zero.
To find the values where the fraction is equal to zero, we solve the equation 1/(x + 6)^2 = 0.
1/(x + 6)^2 = 0
This equation has no real solutions since the numerator is a constant (1) and cannot be equal to zero.
Therefore, the range of the function f(x) = 4 - 1/(x + 6)^2 is (-∞, ∞).
Lastly, let's determine the asymptotes of the function:
1. Vertical asymptote:
A vertical asymptote occurs when the denominator of the function approaches zero. In this case, the denominator is (x + 6)^2. Thus, the vertical asymptote is x = -6.
2. Horizontal asymptote:
To determine the horizontal asymptote, we look at the highest power of x in the function. In this case, the highest power of x is x^0 (since (x + 6)^2 is squared). Thus, the degree of the numerator and denominator is the same, and the horizontal asymptote is given by the quotient of their leading coefficients.
The leading coefficient of the numerator is 4, and the leading coefficient of the denominator is 1. So, the horizontal asymptote is y = 4/1, which simplifies to y = 4.
In summary:
a. The domain of the function is (-∞, -6) U (-6, ∞).
b. The range of the function is (-∞, ∞).
c. The vertical asymptote is x = -6, and the horizontal asymptote is y = 4.