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Consider the function \(f\) with rule \(f(x)=\left\{\begin{array}{cl}\dfrac{x^2+3 x-10}{x-2} & , x \in R \backslash\{2\} \\ 7 & , x=2\end{array}\right.\)
Which of the following statements is correct?
Consider the family of functions \(f\) with rule \(f(x)=\dfrac{x^2}{x-k}\), where \(k \in R \backslash\{0\}\).
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a. \(\text {Asymptotes: } x=1,\ y=x+1\)
b.
c.i. \(\text {Asymptotes: } x=k,\ y=x+k\)
c.ii. \(\text {Distance }=2 \sqrt{5}|k|\)
d.i. \(\displaystyle V=\pi \int_{\frac{-\sqrt{7}-1}{2}}^{\frac{\sqrt{7}-1}{2}}(x+3)^2-\left(\frac{x^2}{x-1}\right)^2 dx\)
d.ii. \(V=51.42\ \text{u}^3 \)
a. \(\text {When } k=1 :\)
\(f(x)=\dfrac{x^2}{x-1}=\dfrac{(x+1)(x-1)+1}{(x-1)}=x+1+\dfrac{1}{x-1}\)
\(\text {Asymptotes: } x=1,\ y=x+1\)
b.
c.i. \(f(x)=\dfrac{x^2}{x-k}=\dfrac{(x+k)(x-k)+k^2}{x-k}=x+k+\dfrac{k^2}{x-k}\)
\(\text {Using part a.}\)
\(\text {Asymptotes: } x=k,\ y=x+k\)
c.ii. \(f^{\prime}(x)=1-\left(\dfrac{k}{x-k}\right)^2\)
\(\text {TP’s when } f^{\prime}(x)=0 \text { (by CAS):}\)
\(\Rightarrow(2 k, 4 k),(0,0)\)
\(\text {Distance }\displaystyle=\sqrt{(2 k-0)^2+(4 k-0)^2}=\sqrt{20 k^2}=2 \sqrt{5}|k|\)
d.i \(\text {Solve for intersection of graphs (by CAS):}\)
\(\displaystyle x+3=\left|\frac{x^2}{x-1}\right|\)
\(\displaystyle \Rightarrow x=\frac{3}{2}, x=\frac{-1 \pm \sqrt{7}}{2}\)
\(\displaystyle V=\pi \int_{\frac{-\sqrt{7}-1}{2}}^{\frac{\sqrt{7}-1}{2}}(x+3)^2-\left(\frac{x^2}{x-1}\right)^2 dx\)
d.ii. \(V=51.42\ \text{u}^3 \text{ (by CAS) }\)
The graph of `y=\frac{x^2+2x+c}{x^2-4}`, where `c \in R`, will always have
`y=\frac{x^2+2x+c}{x^2-4}\ \ =>\ \ y=\frac{x^2+2x+c}{(x-2)(x+2)}`
`text{Vertical asymptotes:}\ x=2, \ x=-2`
`->\ text{However, if}\ c=0,\ \text{only 1 vertical asymptote at}\ \ x=2.`
`text{Horizontal asymptote:}\ y=1`
`text{Using CAS to graph for different values of}\ c:`
`=>E`
The graph of \(y=\dfrac{x^3}{a x^2+b x+c}\) has asymptotes given by \(y=2 x+1\) and \(x=1\). The values of \(a, b\) and \(c\) are, respectively
Consider the function \(f\) with rule \(f(x)=\dfrac{x^2+x-6}{x-1}\).
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| a. | \(f(x)\) | \(=\dfrac{x^2+x-6}{x-1}\) |
| \(=\dfrac{(x-1)(x+2)-4}{x-1}\) | ||
| \(=x+2-\dfrac{4}{x-1} \) |
b. \(\text{Asymptotes at:}\ x=1, \ y=x+2 \)
\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} x \rule[-1ex]{0pt}{0pt} & -3 & -1 & 0 & 1 & 2 \\
\hline
\rule{0pt}{2.5ex} y \rule[-1ex]{0pt}{0pt} & 0 & 3 & 6 & \infty & 0 \\
\hline
\end{array}
| a. | \(f(x)\) | \(=\dfrac{x^2+x-6}{x-1}\) |
| \(=\dfrac{(x-1)(x+2)-4}{x-1}\) | ||
| \(=x+2-\dfrac{4}{x-1} \) |
b. \(\text{Asymptotes at:}\ x=1, \ y=x+2 \)
\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} x \rule[-1ex]{0pt}{0pt} & -3 & -1 & 0 & 1 & 2 \\
\hline
\rule{0pt}{2.5ex} y \rule[-1ex]{0pt}{0pt} & 0 & 3 & 6 & \infty & 0 \\
\hline
\end{array}
The graph of which one of the following relations does not have a vertical asymptote?
The asymptote(s) of the graph of `f(x) = (x^2 + 1)/(2x - 8)` has equation(s)
The number of straight line asymptotes of the graph of `y = (2x^3 + x^2 - 1)/(x^2 - x - 2)` is
A. 0
B. 1
C. 2
D. 3
E. 4
The graph of `y = f(x)` is shown below.
All the axes below have the same scale as the axes in the diagram above.
The graph of `y = 1/(f(x))` is best represented by
| A. |  |
B. |  |
| C. |  |
D. |  |
| E. |  |
The graph with equation `y = 1/(2x^2 - x - 6)` has asymptotes given by
A. `x = -3/2,\ x = 2 and y = 1`
B. `x = -3/2 and x = 2` only
C. `x = 3/2,\ x = -2 and y = 0`
D. `x = -3/2,\ x = 2 and y = 0`
E. `x = 3/2 and x = -2` only
The graph of `y = 1/(ax^2 + bx + c)` has asymptotes at `x = − 5`, `x = 3` and `y = 0`.
Given that the graph has one stationary point with a `y`-coordinate of `−1/8`, it follows that
A. `a = 1`, `b = 2`, `c = −15`
B. `a = 1/2`, `b = −1`, `c = −15/2`
C. `a = −1/2`, `b = −1`, `c = 15`
D. `a = −1`, `b = −2`, `c = −15`
E. `a = 1/2`, `b = 1`, `c = −15/2`
The straight-line asymptote(s) of the graph of the function with rule `f(x) = (x^3 - ax)/x^2`, where `a` is a non-zero real constant, is given by
A. `x = 0` only.
B. `x = 0` and `y = 0` only.
C. `x = 0` and `y = x` only.
D. `x = 0, \ x = sqrt a` and `x = -sqrt a` only.
E. `x = 0` and `y = a` only.
The number of asymptotes of the graph of the function with rule `f(x) = (x^3 - 7x + 5)/(x^2 + 3x - 4)` is
A. 0
B. 1
C. 2
D. 3
E. 4
The features of the graph of the function with rule `f(x) = (x^2 - 4x + 3)/(x^2 - x - 6)` include
A. asymptotes at `x = 1` and `x = −2`
B. asymptotes at `x = 3` and `x = −2`
C. an asymptote at `x = 1` and a point of discontinuity at `x = 3`
D. an asymptote at `x = −2` and a point of discontinuity at `x = 3`
E. an asymptote at `x = 3` and a point of discontinuity at `x = −2`
Sketch the graph of `f(x) = (x + 1)/(x^2 - 4)` on the axes provided below, labelling any asymptotes with their equations and any intercepts with their coordinates. (4 marks)
`text(Using partial fractions:)`
`(x + 1)/(x^2 – 4)= A/(x – 2) + B/(x + 2)`
`A(x + 2) + B (x – 2) = x + 1`
`text(When)\ \ x = 2,`
| `4A` | `= 3 => \ A=3/4` |
`text(When)\ \ x = -2`
| `-4B` | `= -1 =>\ B=1/4` |
`f(x)= 1/(4(x + 2)) + 3/(4(x – 2))`
`text(As)\ \ x->oo, \ f(x)->0^+`
`text(As)\ \ x->- oo, \ f(x)->0^-`
`f(0)= – 1/4`
`text(Find)\ x\ text(when)\ \ f(x)=0:`
| `x + 1` | `= 0` |
| `x` | `= -1` |