Question

Use the provided information to sketch the graph of ff. Assume that ff is continuous on its domain and that all intercepts are included in the table of values.

Domain: All real xx, except x=1;x=-1;

f(3)=2,f(2)=3,f(0)=1,f(1)=0;f(-3)=2,f(-2)=3,f(0)=-1,f(1)=0;

f(x)>0 on (,1) and (1,)f'(x)>0\ \text{on}\ (-\infty,-1)\ \text{and}\ (-1,\infty)

f(x)>0 on (,1);f(x)<0 on (1,);f''(x)>0\ \text{on}\ (-\infty,-1); f''(x)<0\ \text{on}\ (-1,\infty);

vertical asymptote: x=1;x=-1;

horizontal asymptote: y=1y=1

Solution

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Answered 1 year ago
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To sketch the graph of ff, we need to determine the intervals where it is increasing, decreasing, concave upward, concave downward, and the asymptotes.

Note that if f(x)f'(x) is positive then f(x)f(x) is increasing, and if f(x)f'(x) is negative then f(x)f(x) is decreasing.

f(x)f'(x) is positive on (,1)(-\infty,-1). So, f(x)f(x) is increasing on (,2)(-\infty,-2).

f(x)f'(x) is positive on (1,)(-1,\infty). so, f(x)f(x) is increasing on (1,)(-1,\infty).

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