4. For the function defined by
3
2
1
5
x
f x ,
a) determine the equation of the horizontal asymptote.
b) determine the y-intercept.
c) state if the function is increasing or decreasing.
d) state the domain and range.
e) state the transformations in words, and write the corresponding transformation mapping
notation.
f) graph the function.
a) To determine the equation of the horizontal asymptote, we need to look at the behavior of the function as x approaches positive infinity and negative infinity.
As x approaches positive infinity, the term (3x^2 - 5) becomes infinitely large, while the term (2x + 1) remains constant. Therefore, the function approaches positive infinity.
As x approaches negative infinity, the term (3x^2 - 5) becomes infinitely large, while the term (2x + 1) remains constant. Therefore, the function approaches negative infinity.
Since the function approaches both positive and negative infinity as x approaches infinity, there is no horizontal asymptote.
b) To determine the y-intercept, we set x = 0 in the function:
f(0) = (3(0)^2 - 5)/(2(0) + 1) = -5/1 = -5
Therefore, the y-intercept is -5.
c) To determine if the function is increasing or decreasing, we can analyze the behavior of the function using calculus. Taking the derivative of the function:
f'(x) = (6x(2x + 1) - (3x^2 - 5)(2))/(2x + 1)^2
Simplifying the derivative:
f'(x) = (12x^2 + 6x - 6x^2 + 10)/(2x + 1)^2
f'(x) = (6x^2 + 6x + 10)/(2x + 1)^2
The numerator of the derivative, 6x^2 + 6x + 10, is always positive for any value of x. The denominator, (2x + 1)^2, is always positive as well. Therefore, the derivative f'(x) is always positive.
Since the derivative is always positive, the function is increasing.
d) The domain of the function is all real numbers because there are no restrictions on the variable x.
The range of the function is all real numbers because as x approaches infinity, the function approaches positive infinity, and as x approaches negative infinity, the function approaches negative infinity.
e) The function has no transformations, so the transformations in words are "none." The corresponding transformation mapping notation is T(x) = x.
f) The graph of the function is a continuous curve that increases as x increases and decreases as x decreases. It does not have any horizontal asymptotes.