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Solutions
38 Videos

Determine equations for the vertical and horizontal asymptotes of each function.

```
\displaystyle
f(x) = \frac{1}{x-2}
```

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Q1a

Determine equations for the vertical and horizontal asymptotes of each function.

```
\displaystyle
f(x) = \frac{3}{x+7}
```

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Q1b

Determine equations for the vertical and horizontal asymptotes of each function.

```
\displaystyle
f(x) = -\frac{4}{x-7}
```

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Q1c

Determine an equation to represent each function.

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Q2a

Determine an equation to represent each function.

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Q2b

Sketch a graph of each function. State the domain, range, y-intercepts, and equations of the asymptotes.

```
\displaystyle
f(x) = \frac{5}{x-3}
```

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Q3a

Sketch a graph of each function. State the domain, range, y-intercepts, and equations of the asymptotes.

```
\displaystyle
g(x) = -\frac{1}{x-4}
```

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Q3b

Sketch a graph of each function. State the domain, range, y-intercepts, and equations of the asymptotes.

```
\displaystyle
h(x) = \frac{1}{2x-3}
```

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Q3c

```
\displaystyle
h(x) = - \frac{8}{5x+4}
```

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Q3d

Determine equations for the vertical asymptotes of the function. Then, state the domain.

```
\displaystyle
f(x) = \frac{1}{(x -3)(x+ 4)}
```

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Q4a

Determine equations for the vertical asymptotes of the function. Then, state the domain.

```
\displaystyle
f(x) = -\frac{2}{(x+ 3)^2}
```

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Q4b

Determine equations for the vertical asymptotes of the function. Then, state the domain.

```
\displaystyle
f(x) = \frac{1}{x^2 + 8x + 12}
```

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Q4c

,For each function,

- i) determine equations for the asymptotes
- ii) determine the y-intercepts
- iii) sketch a graph
- iv) describe the increasing and decreasing intervals
- v) state the domain and range

```
\displaystyle
f(x) = \frac{1}{x^2 + 6x + 5}
```

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Q5a

,For each function,

- i) determine equations for the asymptotes
- ii) determine the y-intercepts
- iii) sketch a graph
- iv) describe the increasing and decreasing intervals
- v) state the domain and range

```
\displaystyle
f(x) = \frac{1}{x^2 - 5x-24}
```

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Q5b

,For each function,

- i) determine equations for the asymptotes
- ii) determine the y-intercepts
- iii) sketch a graph
- iv) describe the increasing and decreasing intervals
- v) state the domain and range

```
\displaystyle
f(x) = -\frac{1}{x^2 -6x + 9}
```

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Q5c

,For each function,

- i) determine equations for the asymptotes
- ii) determine the y-intercepts
- iii) sketch a graph
- iv) describe the increasing and decreasing intervals
- v) state the domain and range

```
\displaystyle
f(x) = -\frac{2}{x^2+ 5}
```

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Q5d

Analyse the slope, and change in slope, for the
intervals of the function ```
\displaystyle
f(x) = \frac{1}{2x^2 +3x - 5}
```

by sketching a graph of the function.

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Q6

Write an equation for a function that is the reciprocal of a quadratic and has the following properties:

- The horizontal asymptote is
`y = 0`

. - The vertical asymptotes are
`x = -4`

and`x = 5`

- For the intervals
`x < -4`

and`x > 5`

,`y < 0`

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Q7

Determine an equation for the horizontal asymptote of each function.

```
\displaystyle
a(x) = \frac{x}{x + 5}
```

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Q8a

Determine an equation for the horizontal asymptote of each function.

```
\displaystyle
f(x) = -\frac{2x}{x- 3}
```

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Q8b

Determine an equation for the horizontal asymptote of each function.

```
\displaystyle
f(x) = \frac{x + 2}{x-2}
```

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Q8c

Summarize the key features of each function. Then, sketch a graph of the function.

```
\displaystyle
f(x) = \frac{x}{x -2}
```

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Q9a

Summarize the key features of each function. Then, sketch a graph of the function.

```
\displaystyle
f(x) = - \frac{3x}{x + 1}
```

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Q9b

Summarize the key features of each function. Then, sketch a graph of the function.

```
\displaystyle
f(x) = \frac{x-2}{x+ 4}
```

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Q9c

Summarize the key features of each function. Then, sketch a graph of the function.

```
\displaystyle
f(x) = \frac{6x+ 2}{2x-1}
```

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Q9d

Write an equation of a rational function of the
form `f(x) = \frac{ax + b}{cx + d}`

whose graph has all of the
following features:

- x-intercept of
`\frac{1}{4}`

- y-intercept of
`-\frac{1}{2}`

- vertical asymptote with equations
`x = - \frac{2}{3}`

- horizontal asymptote with equations
`x = \frac{4}{3}`

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Q10

Solve algebraically.

```
\displaystyle
\frac{7}{x-4} =2
```

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Q11a

Solve algebraically.

```
\displaystyle
\frac{3}{x^2 + 6x -24} =1
```

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Q11b

You can use a graphing device for this. Solve for x.

```
\displaystyle
\frac{x^2 -3x + 1}{2 -x} =\frac{x^2 + 5x + 4}{x-6}
```

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Q12c

Solve for x.

```
\displaystyle
\frac{3}{x + 5} < 2
```

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Q13a

Solve for x.

```
\displaystyle
\frac{3}{x + 2} \leq \frac{4}{x + 3}
```

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Q13b

Solve for x.

```
\displaystyle
\frac{x^2 -x -20}{x^2 -4x -12} > 0
```

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Q13c

Solve for x.

```
\displaystyle
\frac{x}{x + 5} > \frac{x - 1}{x + 7}
```

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Q13d

Solve for `x`

.

```
\displaystyle
\frac{x^2 + 5x +4}{x^2 -5x + 6} < 0
```

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Q14a

Solve for `x`

.

```
\displaystyle
\frac{x^2 -6x + 9}{2x^2 + 17x + 8} > 0
```

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Q14b

A manufacturer is predicting profit, P, in thousands of dollars. on the sale of x tonnes of fertilizer according to the equation

```
\displaystyle
P(x) = \frac{600x - 15000}{x + 100}
```

a) Sketch a graph of this relation.

b) Describe the predicted profit as sales increase.

c) Compare the rtes of change of the profit at sales of 100 t and 500 t of fertilizer.

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Q15

Sketch a graph of each function. Describe each special case.

```
\displaystyle
f(x) = \frac{x}{x^2 + 5x}
```

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Q16a

Sketch a graph of each function. Describe each special case.

```
\displaystyle
f(x) = \frac{x^2 -2x - 35}{x^2 -3x - 28}
```

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Q16b