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Solutions
31 Videos

For the function, determine the domain and range, intercepts, positive/negative intervals, and increasing and decreasing intervals. Use this information to sketch a graph of the reciprocal function.

`f(x) = 3x + 2`

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Q1a

For the function, determine the domain and range, intercepts, positive/negative intervals, and increasing and decreasing intervals. Use this information to sketch a graph of the reciprocal function.

```
\displaystyle
f(x) = 2x^2 + 7x -4
```

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Q1b

For the function, determine the domain and range, intercepts, positive/negative intervals, and increasing and decreasing intervals. Use this information to sketch a graph of the reciprocal function.

```
\displaystyle
f(x) =2x^2 + 2
```

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Q1c

Given the graphs of f(x) below, sketch the graphs of ```
\displaystyle
\frac{1}{f(x)}
```

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Q2a

Given the graph of f(x) below, sketch the graphs of `y = \frac{1}{f(x)}`

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Q2b

For the function, determine the equations of any vertical asymptotes, the locations of any holes, and the existence of any horizontal or oblique asymptotes.

```
\displaystyle
y = \frac{1}{x+ 17}
```

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Q3a

For the function, determine the equations of any vertical asymptotes, the locations of any holes, and the existence of any horizontal or oblique asymptotes.

```
\displaystyle
y = \frac{2x}{5x+ 3}
```

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Q3b

For the function, determine the equations of any vertical asymptotes, the locations of any holes, and the existence of any horizontal or oblique asymptotes.

```
\displaystyle
y = \frac{3x+ 33}{-4x^2 -42x + 22}
```

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Q3c

```
\displaystyle
y = \frac{3x^2 -2}{x -1}
```

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Q3d

The population of locusts in a Prairie a town over
the last 50 years is modelled by the function
```
\displaystyle
f(x) = \frac{75x}{x^2 + 3x +2}
```

. The locust population is
given in hundreds of thousands. Describe the locust population in the town over time, where x is time in years.

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Q4

For the function, determine the domain, intercepts, asymptotes, and positive/negative intervals. Use these characteristics to sketch the graph of the function. Then describe where the function is increasing or decreasing.

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

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Q5a

For the function, determine the domain, intercepts, asymptotes, and positive/negative intervals. Use these characteristics to sketch the graph of the function. Then describe where the function is increasing or decreasing.

```
\displaystyle
f(x) = \frac{4x -8}{x -2}
```

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0.46mins

Q5b

For the function, determine the domain, intercepts, asymptotes, and positive/negative intervals. Use these characteristics to sketch the graph of the function. Then describe where the function is increasing or decreasing.

```
\displaystyle
f(x) =\frac{x -6}{3x - 18}
```

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0.37mins

Q5c

```
\displaystyle
f(x) = \frac{4x}{2x + 1}
```

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0.28mins

Q5d

Describe how you can determine the behaviour of the values of a rational function on either side of a vertical asymptote.

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Q6

Solve the equation algebraically.

```
\displaystyle
\frac{x-6}{x+ 2} = 0
```

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Q7a

Solve the equation algebraically.

```
\displaystyle
15x + 7 = \frac{2}{x}
```

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0.55mins

Q7b

Solve the equation algebraically.

```
\displaystyle
\frac{2x}{x- 12} = \frac{-2}{x+ 3}
```

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Q7c

Solve the equation algebraically.

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

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0.32mins

Q7d

A group of students have volunteered for the student council car wash. Janet can wash a car in `m`

minutes. Rodriguez can wash a car in
`m- 5`

minutes, while Nick needs the same amount of time as Janet. If they all work together, they can wash a car in about
3.23 minutes. How long does Janet take to wash a car?

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Q8

The concentration of a toxic chemical in a spring-fed lake is given by the equation ```
\displaystyle
c(x) = \frac{50x}{x^2 +3x + 6}
```

, where `c`

is given in grams per litre and `x`

is the time in days. Determine when the concentration of the chemical is `6.16 g/L`

.

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Q9

Solve for `x`

.

```
\displaystyle
-x + 5 < \frac{1}{x + 3}
```

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2.58mins

Q10a

Solve for `x`

.

```
\displaystyle
\frac{55}{3x+ 4} > -x
```

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1.24mins

Q10b

Solve for `x`

.

```
\displaystyle
\frac{2x}{3x +4} > \frac{x}{x + 1}
```

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Q10c

Solve for `x`

.

```
\displaystyle
\frac{x}{6x -9} \leq \frac{1}{x}
```

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Q10d

A biologist predicted that the population of tadpoles in a pond could be modelled by the
function ```
\displaystyle
f(t) = \frac{40t}{t^2 + 1}
```

, where `t`

is given in days. The function that actually models the tadpole population is ```
\displaystyle
g(t) = \frac{45t}{t^2 + 8t + 7}
```

. Determine where `g(t) > f(t)`

.

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Q11

Estimate the slope of the line that is tangent to each function at the given point. At what point(s) is it not possible to draw a tangent line?

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

where `x = 4`

.

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Q12a

Estimate the slope of the line that is tangent to each function at the given point. At what point(s) is it not possible to draw a tangent line?

```
\displaystyle
f(x) = \frac{2x -1}{x^2 +3x + 2}
```

where `x = 1`

.

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Q12b

The concentration, `c`

, of a drug in the bloodstream `t`

hours after the drug was taken orally is given by ```
\displaystyle
c(t) = \frac{5t}{t^2 +7}
```

, where `c`

is measured in milligrams per litre.

a) Calculate the average rate of change in the drug's concentration during the first 2 h since ingestion.

b) Estimate the rate at which the concentration of the drug is changing after exactly 3h.

c) Graph `c(t)`

on a graphing calculator. When is the concentration of the drug increasing the fastest in the bloodstream? Explain.

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Q13

Given the function ```
\displaystyle
f(x) = \frac{2x}{x -4}
```

, determine the coordinates of a point on f(x) where the slope of the tangent line equals the slope of the secant line that passes through `A(5, 10)`

and `B(8, 4)`

.

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Q14

Describe what happens to the slope of a tangent line on the graph of a rational function as the `x`

-coordinate of the point of tangency

a) gets closer and closer to the vertical asymptote.

b) grows larger in both the positive and negative direction.

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Q15