3. For the function defined by
25 6
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 find the equation of the horizontal asymptote, we need to analyze the behavior of the function as x approaches positive infinity and negative infinity. As x approaches positive infinity, the term 5x dominates the -6, so the function approaches positive infinity. Similarly, as x approaches negative infinity, the term 5x dominates the -6, so the function approaches negative infinity. Therefore, there is no horizontal asymptote.
b) The y-intercept is the value of the function when x = 0. Plugging in x = 0, we get f(0) = 2(5(0)) - 6 = -6. Therefore, the y-intercept is -6.
c) To determine if the function is increasing or decreasing, we need to analyze the sign of the derivative. Taking the derivative of the given function, we get f'(x) = 2(5) - 0 = 10. Since the derivative is positive (10 > 0) for all x, the function is increasing.
d) The domain is the set of all real numbers since there are no restrictions on the input x. The range is all real numbers as well since the output f(x) can take any value on the number line.
e) The function has no transformations in terms of shifts or stretches, so there are no corresponding transformation mapping notations.
f) The graph of the function is a straight line passing through the y-intercept (-6) with a positive slope. It increases indefinitely in both the positive and negative x-directions.