Q. 72 - Calculus

Question

Sketch careful, labeled graphs of each function $f$ in Exercises 57–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$ and $f^{\text{'}}$and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right)=\frac{{\left(x-1\right)}^2}{x^2-1}$.

Step-by-Step Solution

Answer

The graph for the function $f\left(x\right)=\frac{{\left(x-1\right)}^2}{x^2-1}$ is,

Step 1 . Given information

$f\left(x\right)=\frac{{\left(x-1\right)}^2}{x^2-1}$.

Step 2 . Let src="data:image/svg+xml;base64,<svg xmlns="http://www.w3.org/2000/svg" xmlns:wrs="http://www.wiris.com/xml/mathml-extension" height="47" width="104" wrs:baseline="29"><!--MathML: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfrac></math>--><defs><style type="text/css">@font-face{font-family:'aec8956637a99787bd197eacd77acce';src:url(data:font/truetype;charset=utf-8;base64,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)format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'math143f4d31b04031e49f5eb18baba';src:url(data:font/truetype;charset=utf-8;base64,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)format('truetype');font-weight:normal;font-style:normal;}@font-face{font-family:'round_brackets18549f92a457f2409';src:url(data:font/truetype;charset=utf-8;base64,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)format('truetype');font-weight:normal;font-style:normal;}</style></defs><text font-family="aec8956637a99787bd197eacd77acce" font-size="16" font-style="italic" text-anchor="middle" x="3.5" y="29">f</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="12.5" y="29">(</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="25.5" y="29">)</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="18.5" y="29">x</text><text font-family="math143f4d31b04031e49f5eb18baba" font-size="16" text-anchor="middle" x="36.5" y="29">=</text><line stroke="#000000" stroke-linecap="square" stroke-width="1" x1="47.5" x2="100.5" y1="23.5" y2="23.5"/><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="52.5" y="18">(</text><text font-family="round_brackets18549f92a457f2409" font-size="16" text-anchor="middle" x="89.5" y="18">)</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="58.5" y="18">x</text><text font-family="math143f4d31b04031e49f5eb18baba" font-size="16" text-anchor="middle" x="70.5" y="18">&#x2212;</text><text font-family="Arial" font-size="16" text-anchor="middle" x="82.5" y="18">1</text><text font-family="Arial" font-size="12" text-anchor="middle" x="95.5" y="11">2</text><text font-family="Arial" font-size="16" font-style="italic" text-anchor="middle" x="58.5" y="43">x</text><text font-family="Arial" font-size="12" text-anchor="middle" x="66.5" y="36">2</text><text font-family="math143f4d31b04031e49f5eb18baba" font-size="16" text-anchor="middle" x="77.5" y="43">&#x2212;</text><text font-family="Arial" font-size="16" text-anchor="middle" x="89.5" y="43">1</text></svg>" localid="1648542375542" f x = x - 1 2 x 2 - 1 .

$$\begin{gathered} f\left(x\right)=\frac{{\left(x-1\right)}^2}{x^2-1} \\ =\frac{{\left(x-1\right)}^2}{\left(x+1\right)\left(x-1\right)} \\ =\frac{\left(x-1\right)}{\left(x+1\right)} \end{gathered}$$

Let $y=\frac{\left(x-1\right)}{\left(x+1\right)}$.

Now point table for the function is given by,

$x$	$y$	$\left(x,y\right)$
$-5$	$\frac{3}{2}$	$\left(-5,\frac{3}{2}\right)$
$-2$	$3$	$\left(-2,3\right)$
$0$	$-1$	$\left(0,-1\right)$
$1$	$0$	$\left(1,0\right)$
$2$	$\frac{1}{3}$	$\left(2,\frac{1}{3}\right)$

Step 3 . The graph for the function is,

Step 4 . Now for critical points, f ' x = 0 .

$$\begin{gathered} \frac{d}{dx}\left(\frac{x-1}{x+1}\right)=0 \\ \frac{\left(x+1\right)\left(1\right)-\left(x-1\right)\left(1\right)}{{\left(x+1\right)}^2}=0 \\ \frac{x+1-x+1}{{\left(x+1\right)}^2}=0 \\ \frac{2}{{\left(x+1\right)}^2}=0 \end{gathered}$$

Therefore, $f$ has no critical points. It has no local extrema.

Step 5 . The sign chart of f is shown below:

For roots of the function,

$$\begin{gathered} \frac{x-1}{x+1}=0 \\ x-1=0 \\ x=1 \end{gathered}$$

Therefore, $x$ has a root which is $1$.

Step 6 . Therefore, the function f is defined everywhere except at x = - 1 and x = 1 .

The root of the function is, $x=1$.

The function is positive on $\left(-\infty ,-1\right)\cup \left(1,\infty \right)$ and negative elsewhere.

Again,

$$\begin{gathered} \underset{x\to -\infty }{\lim }f\left(x\right)=\underset{x\to -\infty }{\lim }\frac{x-1}{x+1} \\ =\infty \\ \underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }\frac{x-1}{x+1} \\ =\infty \end{gathered}$$