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?
REINY OR WHOEVER IS READING THIS, I JUST NEED TO KNOW THE ANSWER FOR D
To answer part D, let's analyze the pattern and then determine if it holds true for all pairs of functions.
The pattern observed in the given functions f(x) and f^r(x) is as follows:
- In f(x), the numerator is x-2 and the denominator is x+4.
- In f^r(x), the numerator is x+4 and the denominator is x-2.
When we interchange the numerators and denominators to find the reciprocal, the pattern remains the same. Only the positions of the numerator and denominator are switched.
Generalizing this pattern, for any pair of functions f(x) = x+b (numerator) / x+d (denominator) and f^r(x) = x+d (numerator) / x+b (denominator), the pattern should continue to hold true.
This is because when we find the reciprocal of any rational function, we simply swap the positions of the numerator and denominator. The properties such as asymptotes, intercepts, symmetry, and intervals of increase/decrease are all preserved.
So, for any pair of functions with the form f(x) = x+b (numerator) / x+d (denominator) and f^r(x) = x+d (numerator) / x+b (denominator), the same pattern observed in parts A, B, and C will occur.