0 votes 0 votes Sketch the graph of $f(x) = \frac{x^{3}-1}{x^{2}-1}$ Calculus calculus + – kidussss asked Jul 29, 2022 kidussss 363 views answer comment Share Follow See all 6 Comments See all 6 6 Comments reply Lakshman Bhaiya commented Jul 29, 2022 reply Follow Share @kidussss Please mention the source of the questions. 0 votes 0 votes kidussss commented Jul 29, 2022 reply Follow Share This Mathematics for Natural science students.https://gateoverflow.in/?qa=blob&qa_blobid=8154948231667306571 0 votes 0 votes ankitgupta.1729 commented Jul 29, 2022 reply Follow Share Following tools might help you to draw the graph for the given function, $f(x) = \frac{x^3 – 1}{x^2 – 1}$ Here, $f(x)$ is undefined for $\pm 1.$ Hence domain for the given function is $(-\infty,-1) \cup (-1,1) \cup (1,\infty)$ Now, we try to find out for what interval, function is increasing or decreasing and for this, sign of the derivative of $f(x)$ would help us. $f’(x) = \frac{x(x+2)}{(x+1)^2}$ here denominator is always $>0$ for the given domain. And so, sign of $f’(x)$ is decided by $x(x+2).$ Here, $f’(x)$ is negative for interval $(-2,0)$ and positive for $(-\infty,-2) \cup (0,\infty)$ (You can verify it by making parabola for $x(x+2)$ and check when it is above x axis and when it is below. It means $f(x)$ increases for the interval $(-\infty,-2) \cup (0,\infty)$ and decreases for the interval $(-2,0)$ Now, try to find out asymptotes. $\lim_{x \rightarrow -1^+} \frac{x^3 - 1}{x^2 -1} = \infty$ and $\lim_{x \rightarrow -1^-} \frac{x^3 - 1}{x^2 -1} = -\infty$ $\lim_{x \rightarrow \infty} \frac{x^3 - 1}{x^2 -1} = \infty$ $\lim_{x \rightarrow -\infty} \frac{x^3 - 1}{x^2 -1} = -\infty$ Now, here, $f’(x) = \frac{x(x+2)}{(x+1)^2}$ which has roots for $x=0,-2$ and so, local minima is for $x=0$ and local maxima is for $x=-2$ and $f(0)=1$ and $f(-2) = -3$ and hence, range for $f(x)$ is $(-\infty, -3] \cup [1,\infty)$ You might check concavity also and can try other things also but if you follow the above points, you plot the given function $f(x)$ which you can verify here 1 votes 1 votes Hailemariam commented Aug 3, 2022 reply Follow Share Please show with skech graph. I’m not understand it. 0 votes 0 votes ankitgupta.1729 commented Aug 3, 2022 reply Follow Share click here This is the link for the graph of the given function. To sketch manually, you have to follow the points mentioned above. 1 votes 1 votes Hailemariam commented Aug 3, 2022 reply Follow Share Tnks a lot 1 votes 1 votes Please log in or register to add a comment.