f(x) is a linear function which is increasing for XER.
a) Will the reciprocal function be increasing or decreasing?
b)Will there ever be any horizontal asymptotes for the function? If so, where?
c)Will there ever be any horizontal asmptotes for the reciprocal function? If so, where?
d)Will there ever be any vertical asmptotes for the function? If so, where?
e)Will there ever be any vertical asmptotes for the reciprocal function? If so, where?
Thanks, it doesn't matter what equation I use, correct?
not really, as long as you have a linear equation of the form
y = mx + b, with m a positive number (positive slope)
Use relatively small numbers so you don't run off your graph too fast.
a) To determine whether the reciprocal function of f(x) is increasing or decreasing, we need to understand the relationship between the original function and its reciprocal.
The reciprocal of a number x is 1/x, and in general, when a function f(x) increases, its reciprocal 1/f(x) decreases. Similarly, when f(x) decreases, 1/f(x) increases.
Since f(x) is a linear function that is increasing for all real values of x, its reciprocal will be a decreasing function.
b) A horizontal asymptote is a horizontal line that a function approaches as x approaches positive or negative infinity. For a linear function f(x), there will not be any horizontal asymptotes because its graph will extend indefinitely in both the positive and negative directions.
c) For the reciprocal function of f(x), there may be horizontal asymptotes depending on the behavior of f(x). If the linear function f(x) is increasing and approaches a non-zero value as x approaches positive or negative infinity, then the reciprocal function 1/f(x) will approach a horizontal asymptote of y = 0. However, if f(x) approaches zero as x approaches positive or negative infinity, then the reciprocal function will not have a horizontal asymptote.
d) A vertical asymptote is a vertical line that a function approaches as x approaches a certain value. Since f(x) is a linear function, it will not have any vertical asymptotes. Linear functions do not exhibit behaviors like approaching infinity or undefined values as x approaches specific points.
e) The reciprocal function of f(x) may have vertical asymptotes depending on the behavior of f(x). If f(x) has any x-values where it approaches zero, then the reciprocal function 1/f(x) will have vertical asymptotes at those x-values. Vertical asymptotes for the reciprocal function occur when the original function approaches zero, causing the reciprocal to approach infinity or negative infinity. However, if f(x) never approaches zero, then the reciprocal function will not have any vertical asymptotes.
I suggest you take an increasing linear function such as
f(x) = 2x+1 and form its reciprocal , g(x) = 1/(2x+1)
Investigate by graphing each one using graph paper.