1. determine the interval(s) where the function f(x)= -1 / 2x+10 is
a) positive
b) Increasing
2. Consider the function f(x) = 3 / 4x-5
a) Determine the key features of the function:
i) Domain and range
ii)Intercepts
iii) Equations of any asymptotes
iv) Intervals where the functions are increasing and intervals where the function is decreasing
b) Sketch a graph of the function
so, which parts give you trouble?
what do you have so far?
I don't understand both questions
To determine the intervals where a function is positive or increasing, we need to analyze the behavior of the function.
1. Determining the intervals where the function f(x) = -1 / (2x + 10) is positive:
To find the intervals where the function is positive, we need to identify the values of x that make the function greater than zero (positive).
We can start by setting the function equal to zero and solving for x:
-1 / (2x + 10) > 0
Solving for x, we get:
2x + 10 < 0
From this, we can determine that x < -5.
However, we also need to consider that the denominator (2x + 10) cannot be equal to zero or negative. Therefore, we exclude the value -5 from the solution.
So, the interval where the function f(x) is positive is x < -5.
2. Determining the intervals where the function f(x) = 3 / (4x - 5) is increasing:
To find the intervals where the function is increasing, we need to examine the slope of the function. In this case, the slope is the derivative of the function.
Differentiating the function f(x) = 3 / (4x - 5) with respect to x, we get:
f'(x) = d/dx (3 / (4x - 5)) = -12 / (4x - 5)^2
To determine where the derivative is positive (indicating an increasing function), we set the derivative greater than zero:
-12 / (4x - 5)^2 > 0
Since the denominator cannot be equal to zero, we exclude the value 5/4 from the solution.
Solving further, we get:
-12 > 0
Since -12 is negative, we see that whenever the denominator is positive (4x - 5 > 0), the derivative (slope) is negative. Therefore, the function is decreasing in this interval.
By taking the domain of the function into account (all real numbers except x = 5/4), we conclude that the intervals where the function is increasing are:
(-∞, 5/4) ∪ (5/4, +∞).
Now, moving on to the second question:
a) Key features of the function f(x) = 3 / (4x - 5):
i) Domain and range:
The domain of the function f(x) is all real numbers except x = 5/4 (since the denominator cannot be equal to zero). The range of the function is all real numbers.
ii) Intercepts:
To find the x-intercept (where the function crosses the x-axis), we set f(x) equal to zero:
3 / (4x - 5) = 0
Solving for x, we get:
4x - 5 = 0
4x = 5
x = 5/4
Therefore, the x-intercept is x = 5/4.
To find the y-intercept (where the function crosses the y-axis), we set x equal to zero:
f(0) = 3 / (4(0) - 5) = -3/5
Therefore, the y-intercept is y = -3/5.
iii) Equations of any asymptotes:
This function has a horizontal asymptote. To find it, we analyze the limit as x approaches positive or negative infinity:
lim(x -> ±∞) f(x) = lim(x -> ±∞) (3 / (4x - 5)) = 0
Therefore, the equation of the horizontal asymptote is y = 0.
iv) Intervals where the function is increasing and decreasing:
We already determined this in the first question - the function is increasing on the interval (-∞, 5/4) ∪ (5/4, +∞) and decreasing on the interval (5/4, 25/4).
b) To sketch a graph of the function, we take into account the features we just determined:
- Plot the x-intercept at (5/4, 0)
- Plot the y-intercept at (0, -3/5)
- Draw a horizontal asymptote at y = 0
- Indicate that the function is increasing on (-∞, 5/4) ∪ (5/4, +∞) and decreasing on (5/4, 25/4)
The graph should resemble a hyperbola that approaches the horizontal asymptote as x tends toward positive or negative infinity.