use the asymptotes and intercepts to make a quick sketch of the function f(x)= x-2 (numerator) x+4(denominator) and its reciprocal f^r = x+4 (numerator) x-2 (denominator) on the same set of axes.
b) describe the symmetry in the graphs in part a)
c) determine the intervals of increase and decrease for both f and f^r, how do the sets of intervals compare?
d) would the pattern occur for all pairs of functions f(x)= x+b (numerator) x+d and f^r(x) = x+d (numerator)and x+b (denominator)? explain why or why not?
James, I see 6 posts done by you of this same question.
Twice I have answered it. In one of them I pointed out how you should type your functions, in another I gave you a solution.
Here is my last reply to your question
http://www.jiskha.com/display.cgi?id=1364865245
To make a quick sketch of the function f(x) = (x - 2) / (x + 4) and its reciprocal f^r = (x + 4) / (x - 2), we can use the asymptotes and intercepts.
a) Asymptotes and Intercepts:
1. Horizontal Asymptotes:
The horizontal asymptote for f(x) can be found by looking at the degrees of the numerator and denominator. Since the degree of the numerator is 1 and the degree of the denominator is also 1, the horizontal asymptote is y = 1. This means that as x approaches positive or negative infinity, the function will approach y = 1.
2. Vertical Asymptotes:
To find the vertical asymptote for f(x), we set the denominator equal to zero and solve for x. In this case, we have (x + 4) = 0, so x = -4. Thus, we have a vertical asymptote at x = -4.
3. x-Intercept:
To find the x-intercept, we set the numerator equal to zero and solve for x. In this case, we have (x - 2) = 0, so x = 2. Thus, we have an x-intercept at x = 2.
4. y-Intercept:
To find the y-intercept, we set x equal to zero and evaluate the function. In this case, we have f(0) = (0 - 2) / (0 + 4) = -1/2. Thus, we have a y-intercept at y = -1/2.
b) Symmetry:
The function f(x) = (x - 2) / (x + 4) does not exhibit any symmetry. However, the reciprocal function f^r(x) = (x + 4) / (x - 2) does exhibit symmetry about the line y = x. This means that the graphs of f(x) and f^r(x) will be mirror images of each other when reflected about the line y = x.
c) Intervals of Increase and Decrease:
To determine the intervals of increase and decrease, we need to analyze the sign of the derivative of the functions.
For f(x), we find the derivative:
f'(x) = (x + 4 - (x - 2)) / (x + 4)^2 = 6 / (x + 4)^2
The sign of the derivative, f'(x), tells us whether the function is increasing or decreasing:
- When f'(x) > 0, the function is increasing.
- When f'(x) < 0, the function is decreasing.
Since 6 is positive, f'(x) is always positive. Therefore, f(x) is always increasing. The intervals of increase for f(x) are (-∞, ∞).
For f^r(x), we find the derivative:
f^r'(x) = (x - 2 - (x + 4)) / (x - 2)^2 = -6 / (x - 2)^2
The sign of the derivative, f^r'(x), tells us whether the function is increasing or decreasing:
- When f^r'(x) > 0, the function is increasing.
- When f^r'(x) < 0, the function is decreasing.
Since -6 is negative, f^r'(x) is always negative. Therefore, f^r(x) is always decreasing. The intervals of decrease for f^r(x) are (-∞, ∞).
Comparing the sets of intervals, we see that f(x) is always increasing while f^r(x) is always decreasing. They are complementary in terms of intervals of increase and decrease.
d) Would the pattern occur for all pairs of functions f(x) = x + b (numerator) / (x + d) and f^r(x) = (x + d) (numerator) / (x + b) ?
Yes, the pattern of one function being always increasing while the reciprocal function being always decreasing would occur for all pairs of functions of the given form.
This pattern is due to the fact that the denominator is being subtracted in the numerator for both f(x) and f^r(x), while the numerator is being added in the denominator for f(x) and f^r(x). This opposite operation causes the function to behave differently, resulting in one always increasing and the other always decreasing.