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?

Did you even look at my answer before you reposted ?

http://www.jiskha.com/display.cgi?id=1364855005

Function:f(x)=x+4

x-2
x-intercept?
y-intercept?
Horizontal Asymptote?
Vertical Asymptote?
Oblique Asymptote?

a) To sketch the function f(x) = (x-2)/(x+4) and its reciprocal f^r(x) = (x+4)/(x-2), we can start by analyzing the asymptotes and intercepts.

1) Asymptotes: The vertical asymptote is found by setting the denominator equal to zero and solving for x. In this case, x + 4 = 0, so x = -4. This gives us a vertical asymptote at x = -4.

The horizontal asymptote is determined by comparing the degrees of the numerator and denominator. In this case, the degrees are both 1, so the horizontal asymptote is y = 1/1 = 1.

2) Intercepts: The x-intercept is found by setting the numerator equal to zero and solving for x. In this case, x - 2 = 0, so x = 2. This gives us an x-intercept at x = 2.

The y-intercept is found by evaluating the function at x = 0. Plugging in x = 0 into f(x), we get f(0) = (-2)/(4) = -1/2. So the y-intercept is at y = -1/2.

Now, we can sketch the graph of f(x) = (x-2)/(x+4) and f^r(x) = (x+4)/(x-2) on the same set of axes, using the asymptotes and intercepts as guides.

b) In part a), the graphs of f(x) and f^r(x) are NOT symmetric to the y-axis. This is because the function f(x) has a vertical asymptote at x = -4, whereas the reciprocal function f^r(x) does not.

c) To determine the intervals of increase and decrease, we need to analyze the signs of the derivatives of f(x) and f^r(x).

For f(x):
- The function f(x) is increasing where f'(x) > 0.
- The function f(x) is decreasing where f'(x) < 0.

To find f'(x), we need to differentiate f(x) using the quotient rule. This gives us:
f'(x) = [(x+4)(1) - (x-2)(1)] / (x+4)^2 = 6 / (x+4)^2

Since the derivative f'(x) is always positive (6>0), this means f(x) is always increasing for all x.

For f^r(x):
- The function f^r(x) is increasing where f^r'(x) > 0.
- The function f^r(x) is decreasing where f^r'(x) < 0.

To find f^r'(x), we need to differentiate f^r(x) using the quotient rule. This gives us:
f^r'(x) = [(x-2)(1) - (x+4)(1)] / (x-2)^2 = -6 / (x-2)^2

Since the derivative f^r'(x) is always negative (-6<0), this means f^r(x) is always decreasing for all x.

Comparing the intervals of increase and decrease for f(x) and f^r(x), we see that while f(x) is always increasing, f^r(x) is always decreasing.

d) The pattern would not occur for all pairs of functions f(x) = x + b (numerator) and f^r(x) = x + d (numerator) / (x + b) (denominator).

The reason is that the pattern is specific to the given functions, where the numerators and denominators are swapped in the reciprocal. In general, the intervals of increase and decrease depend on the specific forms of the functions and their derivatives. Therefore, we cannot assume that the pattern will hold for all pairs of functions.