f(x)=(x+2)/(1-x) (a) Find the domain of f(x). (b) Find the x and y intercepts of
ID: 3193727 • Letter: F
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f(x)=(x+2)/(1-x) (a) Find the domain of f(x). (b) Find the x and y intercepts of the graph y = f(x). (c) Find all vertical asymptotes of f(x). For each vertical asymptote x = a, find the right and left limit (d) Find all horizontal asymptotes of f(x). (e) Find the intervals where f(x) is decreasing, and the intervals where f(x) is increasing (note that f0 may change sign at critical numbers and at discontinuities of f). (f) Find the intervals where f(x) is concave up, and the intervals where f(x) is concave down (note that f00 may change sign at discontinuities of f). (g) Use the information found in parts (a) to (f) to sketch the graph of y = f(x).Explanation / Answer
a) when x=1 f(x) is undefined so the domain of f(x) is all the real numbers except for x=1 or x?1. The only value that f(x) cannot take on is 0. Therefore the range of f(x) is all the real numbers except for x=0 or x?0. b) If f(x) = 1/(1-x) the composition f(f(x)) is just found by substituting f(x) for x so f(f(x) = 1/[1-f(x)] and so f(f(x)) = 1/[1-1/(1-x)] Let g(x) = f(f(x)) g(x) = 1/[(1-x)/(1-x) - 1/(1-x)] = 1/[-x/(1-x)] = (x-1)/x = 1-1/x As 1/x is undefined the domain of this function is all the real numbers except x=0 c) h(x) = g(f(x) = f(f(f(x))) = 1-1/[1/(1-x)] = 1 - (1-x) = x The domain of this function is the entire real number line. d) h(x) = x is just the line y=x
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