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Solutions
122 Videos

a) What is meant by even symmetry? odd symmetry?

b) Describe how to determine whether a function has even or odd symmetry.

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Q1

Describe two key differences between polynomial functions and non-polynomial functions.

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Q2

Compare the end behaviour of the following functions. Explain any differences.

```
\displaystyle
f(x) =-3x^2
```

```
\displaystyle
g(x) =5x^4
```

```
\displaystyle
h(x) = 0.5x^3
```

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Q3

Determine the degree of the polynomial function modelling the following data.

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Q4

Determine an equation and sketch a graph of the function with a base function of `f(x) = x^4`

that has been transformed by `-2f(x - 3) + 1`

.

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Q5

Sketch the functions `f(x) =x^3`

and `g(x) = -\frac{1}{2}(x-1)(x +2)^2`

on the same set of axes. Label the x- and y-intercepts. State the domain and range of each function.

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Q6

Consider the function

```
\displaystyle
f(x) = 2x^4 + 5x^3 -x^2 -3x + 1
```

.

a) Determine the average slope between the points where `x = 1`

and `x = 3`

.

b) Determine the instantaneous slope at each of these points.

c) Compare the three slopes and describe how the graph is changing.

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Q7

Determine the equation of the function.

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Q8a

Determine the equation of the function.

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Q8b

Given the function `f(x) = -2x^2 + 1`

, describe
the slope and the change in slope for the appropriate intervals.

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Q9

Perform each division. Write the statement that can be used to check each division. State the restrictions.

```
\displaystyle
(4x^3 + 6x^2 -4x + 2) \div (2x -1)
```

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Q10a

Perform each division. Write the statement that can be used to check each division. State the restrictions.

```
\displaystyle
(2x^3 -4x +8)\div (x -2)
```

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Q10b

Perform each division. Write the statement that can be used to check each division. State the restrictions.

```
\displaystyle
(x^3 -3x^2 + 5x -4)\div (x + 2)
```

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Q10c

```
\displaystyle
(5x^4 -3x^3 + 2x^2 + x -6)\div (x + 1)
```

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Q10d

Factor if possible.

```
\displaystyle
x^3 + 4x^2 +x -6
```

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Q11a

Factor if possible.

```
\displaystyle
2x^3 + x^2 -16x -16
```

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Q11b

Factor if possible.

```
\displaystyle
x^3 - 7x^2 + 11 x -2
```

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Q11c

Factor if possible.

```
\displaystyle
x^4 + x^2 + 1
```

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Q11d

Use the remainder theorem to determine the remainder.

`4x^3 -7x^2 + 3x + 5`

divided by `x -5`

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Q12a

Use the remainder theorem to determine the remainder.

`6x^4 +7x^2 - 2x -4`

divided by `3x +2`

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Q12b

Use the factor theorem to determine whether the second polynomial is a factor of the first.

```
\displaystyle
3x^5-4x^3 -4x^2 +15
```

, ```
\displaystyle
x + 5
```

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Q13a

Use the factor theorem to determine whether the second polynomial is a factor of the first.

```
\displaystyle
2x^3 -4x^2 + 6x + 5
```

, ```
\displaystyle
x +1
```

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Q13b

Solve for x.

```
\displaystyle
x^4-81 = 0
```

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Q14a

Solve for x.

```
\displaystyle
x^3-x^2-10x - 8=0
```

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Q14b

Solve.

```
\displaystyle
8x^3 + 27 = 0
```

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Q14c

Solve.

```
\displaystyle
12x^4-7x^2 -6x + 16x^3 = 0
```

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Q14d

A family of quartic functions has zeros `-3, -1`

, and `1`

(order 2).

a) Write an equation for the family. State two other members of the family.

b) Determine an equation for the member of the family that passes through the point (-2, 6).

c) Sketch the function you found in part b).

d) Determine the intervals where the function

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Q15

Solve each inequality, showing the appropriate steps. Illustrate your solution on a number line.

```
\displaystyle
(x - 4)(x + 3) > 0
```

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Q16a

Solve each inequality, showing the appropriate steps. Illustrate your solution on a number line.

```
\displaystyle
2x^2 + x- 6 < 0
```

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Q16b

Solve each inequality, showing the appropriate steps. Illustrate your solution on a number line.

```
\displaystyle
x^3 -2x^2 -13x \leq 10
```

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Q16c

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

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

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Q17a

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

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

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Q17b

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

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

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Q17c

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

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

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Q17d

For each function,

i) determine equations for the asymptotes

ii) determine the intercepts

iii) sketch a graph

Iv) describe the increasing intervals and the decreasing intervals

v) state the domain and the range

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

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Q18a

For each function,

i) determine equations for the asymptotes

ii) determine the intercepts

iii) sketch a graph

Iv) describe the increasing intervals and the decreasing intervals

v) state the domain and the range

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

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Q18b

For each function,

i) determine equations for the asymptotes

ii) determine the intercepts

iii) sketch a graph

Iv) describe the increasing intervals and the decreasing intervals

v) state the domain and the range

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

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Q18c

For each function,

i) determine equations for the asymptotes

ii) determine the intercepts

iii) sketch a graph

Iv) describe the increasing intervals and the decreasing intervals

v) state the domain and the range

```
\displaystyle
h(x) = \frac{2x + 3}{5x + 1}
```

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Q18d

For each function,

i) determine equations for the asymptotes

ii) determine the intercepts

iii) sketch a graph

Iv) describe the increasing intervals and the decreasing intervals

v) state the domain and the range

```
\displaystyle
h(x) = \frac{10}{x^2}
```

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Q18e

For each function,

i) determine equations for the asymptotes

ii) determine the intercepts

iii) sketch a graph

Iv) describe the increasing intervals and the decreasing intervals

v) state the domain and the range

```
\displaystyle
h(x) = \frac{3}{x^2 - 6x -27}
```

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Q18f

Analyse the slope and the change in slope for the appropriate intervals of the function ```
\displaystyle
f(x) = \frac{1}{x^2 -4x -21}
```

. Sketch a graph of the function.

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Q19

Solve algebraically.

```
\displaystyle
\frac{5}{x -3} =4
```

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Q20a

Solve algebraically.

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

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Q20b

Solve algebraically.

```
\displaystyle
\frac{}{x^2 + 4x + 7} = 2
```

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Q20c

Solve each inequality. Illustrate the solution on a number line.

```
\displaystyle
\frac{3}{x -4} < 5
```

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Q21a

Solve each inequality. Illustrate the solution on a number line.

```
\displaystyle
\frac{x^2 -8x + 15}{x^2 + 5x + 4} \geq 0
```

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Q21b

A lab technician pours a quantity of a chemical into a beaker of water. The rate, R, in grams per second, at which the chemical dissolves can be modelled by the function ```
\displaystyle
R(t) = \frac{2t}{t^2 + 4t}
```

, where t is the time, in seconds.

a) By hand or using technology, sketch a graph of this relation.

b) What is the equation of the horizontal asymptote? What is its significance?

c) State an appropriate domain for this relation if a rate of 0.05 g/s or less is considered to be inconsequential.

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Q22

Determine the exact radian measure for each angle.

```
\displaystyle
135^o
```

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Q23a

Determine the exact radian measure for each angle.

```
\displaystyle
-60^o
```

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Q23b

Determine the exact degree measure for each angle.

a) ```
\displaystyle
\frac{\pi}{6}
```

b) ```
\displaystyle
\frac{9\pi}{8}
```

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Q24

A sector angle of a circle with radius 9 cm measures ```
\displaystyle
\frac{5\pi}{12}
```

. What is the perimeter of the sector?

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Q25

Determine the exact value of each trigonometric ratio.

```
\displaystyle
\cos \frac{5\pi}{6}
```

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Q26a

Determine the exact value of each trigonometric ratio.

```
\displaystyle
\sin \frac{3\pi}{2}
```

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Q26b

Determine the exact value of each trigonometric ratio.

```
\displaystyle
\tan \frac{4\pi}{3}
```

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Q26c

Determine the exact value of each trigonometric ratio.

```
\displaystyle
\cot \frac{11\pi}{4}
```

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Q26d

Use the sum or difference formulas to find the exact value of each.

```
\displaystyle
\cos \frac{\pi}{12}
```

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Q27a

Use the sum or difference formulas to find the exact value of each.

```
\displaystyle
\sin \frac{11\pi}{12}
```

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Q27b

Prove each identity.

```
\displaystyle
\sec x - \tan x = \frac{1 - \sin x}{\cos x}
```

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Q28a

Prove each identity.

```
\displaystyle
(\csc x - \cot x)^2 = \frac{1- \cos x}{1 + \cos x}
```

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Q28b

Prove each identity.

```
\displaystyle
\sin 2A = \frac{2\tan A}{\sec^2A}
```

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Q28c

Prove each identity.

```
\displaystyle
\cos(x + y)\cos(x - y) = \cos^2x + \cos^2y - 1
```

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Q28d

Given `\sin x = \frac{1}{5}`

and `\sin y = \frac{5}{6}`

, where `x`

and `y`

are acute angles, determine the exact value of `\sin(x + y)`

.

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Q29

Given that `\cos \frac{5\pi}{8} = -\sin y`

, first express `\frac{5\pi}{8}`

as a sum of `\frac{\pi}{2}`

and an angle, and then apply a trigonometric identity to determine the measure of angle `y`

.

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Q30

a) State the period, amplitude, phase shift, and vertical translation for the function ```
\displaystyle
f(x) = 3\sin[2(x - \frac{\pi}{2})] + 4
```

b) State the domain and the range of `f(x)`

.

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Q31

Sketch a graph of each function for one period. Label the x—intercepts and any asymptotes.

```
\displaystyle
f(x) = \sin(x - \pi) -1
```

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Q32a

Sketch a graph of each function for one period. Label the x—intercepts and any asymptotes.

```
\displaystyle
f(x) = -3\cos[4(x + \frac{\pi}{2})]
```

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Q32b

Sketch a graph of each function for one period. Label the x—intercepts and any asymptotes.

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

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Q32c

Solve for `\theta \in [0, 2\pi]`

```
\displaystyle
2\sin \theta = - \sqrt{3}
```

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Q33a

Solve for `\theta \in [0, 2\pi]`

```
\displaystyle
2\sin \theta \cos \theta - \cos \theta = 0
```

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Q33b

Solve for `\theta \in [0, 2\pi]`

```
\displaystyle
\csc^2\theta = 2 + \csc \theta
```

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Q33c

The blade of a sabre saw moves up and down. Its vertical displacement in the first cycle is shown in the table.

a) Make a scatter plot of the data.

b) Write a sine function to model the data.

c) Graph the sine function on the same set of axes as in part a).

d) Estimate the rate of change when the displacement is 0 cm, to one decimal place.

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Q34

Express in logarithmic form.

a) ```
\displaystyle
7^2 =49
```

b) ```
\displaystyle
a^b = c
```

c) ```
\displaystyle
8^3 = 512
```

d) ```
\displaystyle
11^x = y
```

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Q35

a) Sketch graphs of `f(x) = \log x`

and
`g(x) = \frac{1}{2}\log(x+ 1)`

on the same set of
axes. Label the interce pts and any asymptotes.

b) State the domain and the range of each function.

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Q36

Express in exponential form.

a) ```
\displaystyle
\log_36561 = 8
```

b) ```
\displaystyle
\log_a75 = b
```

c) ```
\displaystyle
\log_72401 = 4
```

d) ```
\displaystyle
\log_a19 = b
```

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Q37

Evaluate.

```
\displaystyle
\log_2256
```

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Q38a

Evaluate.

```
\displaystyle
\log_{15} 15
```

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Q38b

Evaluate.

```
\displaystyle
\log_{6} \sqrt{6}
```

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Q38c

Evaluate.

```
\displaystyle
\log_{3} 243
```

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Q38d

Evaluate.

```
\displaystyle
\log_{12} 12
```

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Q38e

Evaluate.

```
\displaystyle
\log_{11} \frac{1}{\sqrt{121}}
```

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Q38f

Solve for x.

```
\displaystyle
\log_3x =4
```

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Q39a

Solve for x.

```
\displaystyle
\log_x125 =3
```

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Q39b

Solve for x.

```
\displaystyle
\log_7x =5
```

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Q39c

Solve for x.

```
\displaystyle
\log_x729 =6
```

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Q39d

Solve for x.

```
\displaystyle
\log_{\frac{1}{2}}128 =x
```

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Q39e

Solve for x.

```
\displaystyle
\log_{\frac{1}{4}}64 = x
```

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Q39f

A culture begins with 100 000 bacteria and grows to 125 000 bacteria after 2.0 min. What is the doubling period, to the nearest minute?

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Q40

The pH scale is defined as pH = —log[`H^+`

], where [`H^+`

] is the concentration of hydronium ions, in moles per litre.

a) Eggs have a pH of 7.8. Are eggs acidic or alkaline? What is the concentration of hydronium ions in eggs?

b) A weak vinegar solution has a hydronium ion concentration of `7.9 X 10^{-4}`

mol/L. What is the pH of the solution?

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Q41

Solve each equation. Check for extraneous roots.

```
\displaystyle
3^{2x } + 3^x - 21 = 0
```

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Q42a

Solve each equation. Check for extraneous roots.

```
\displaystyle
4^{x } + 15(4^{-x}) = 8
```

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Q42b

Use the laws of logarithms to evaluate.

```
\displaystyle
\log_84 + \log_8128
```

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Q43a

Use the laws of logarithms to evaluate.

```
\displaystyle
\log_7 7\sqrt{7}
```

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Q43b

Use the laws of logarithms to evaluate.

```
\displaystyle
\log_5 10 -\log_5250
```

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Q43c

Use the laws of logarithms to evaluate.

```
\displaystyle
\log_6\sqrt[3]{6}
```

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Q43d

Solve, correct to four decimal places.

a) ```
\displaystyle
2^x = 13
```

b) ```
\displaystyle
5^{2x+1} = 97
```

c) ```
\displaystyle
3^x = 19
```

d) ```
\displaystyle
4^{3x+2} = 18
```

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Q44

Solve.

```
\displaystyle
\log_5(x+ 2) + \log_5(2x-1) =2
```

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Q45a

Solve.

```
\displaystyle
\log_4(x+ 3) + \log_4(x + 4) =\frac{1}{2}
```

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Q45b

Determine the point(s) of intersection of the functions `f(x) = \log x`

and `g(x) = \frac{1}{2}\log(x + 1)`

.

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Q46

Bismuth is used in making chemical alloys, medicine, and transistors. A 10-mg sample of bismuth—214 decays to 9 mg in 3 min.

a) Determine the half-life of bismuth-214.

b) How much bismuth-214 remains after 10 min?

c) Graph the amount of bismuth-214 remaining as a function of time.

d) Describe how the graph would change if the half-life were shorter. Give reasons for your answer.

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Q47

The volume of computer parts in landfill sites is growing exponentially. In 2001, a particular landfill site had 124 000 m3 of computer parts, and in 2007, it had 347 000 m3 of parts.

a) What is the doubling time of the volume of computer parts in this landfill site?

b) What is the expected volume of computer parts in this landfill site in 2020?

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Q48

The value of a particular model of car depreciates by 18% per year. This model of car sells for $35 000.

a) Write an equation to relate the value of the car to the time, in years.

b) Determine the value of the car after 5 years.

c) How long will it take for the car to depreciate to half its original value?

d) Sketch a graph of this relation.

e) Describe how the shape of the graph would change if the rate of depreciation changed to 25%.

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Q49

Consider `f(x) = 2^{- \frac{x}{\pi}}`

and `g(x) = 2\cos(4x)`

for `x \in [0, 4\pi]`

. Sketch a graph of each function.

```
\displaystyle
y = f(x) + g(x)
```

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Q50a

Given `f(x) = 2x^2 + 3x - 5`

and `g(x) =x + 3`

, determine

```
\displaystyle
f(g(x))
```

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Q51a

Given `f(x) = 2x^2 + 3x - 5`

and `g(x) =x + 3`

, determine

```
\displaystyle
g(f(x))
```

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Q51b

Given `f(x) = 2x^2 + 3x - 5`

and `g(x) =x + 3`

, determine

```
\displaystyle
f(g(-3))
```

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Q51c

Given `f(x) = 2x^2 + 3x - 5`

and `g(x) =x + 3`

, determine

```
\displaystyle
g(f(7))
```

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Q51d

If `f(x) = \frac{1}{x}`

and `g(x) = 4 -x`

, determine the following

```
\displaystyle
f(g(3))
```

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Q52a

If `f(x) = \frac{1}{x}`

and `g(x) = 4 -x`

, determine the following

```
\displaystyle
f(g(0))
```

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Q52b

If `f(x) = \frac{1}{x}`

and `g(x) = 4 -x`

, determine the following

```
\displaystyle
f(g(4))
```

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Q52c

If `f(x) = \frac{1}{x}`

and `g(x) = 4 -x`

, determine the following

```
\displaystyle
g(f(4))
```

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Q52d

Find expressions for f(g(x)) and g(f(x)), and state their domains.

```
\displaystyle
f(x)= \sqrt{x}
```

, ```
\displaystyle
g(x) = x + 1
```

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Q53a

Find expressions for f(g(x)) and g(f(x)), and state their domains.

```
\displaystyle
f(x)= \sin x
```

, ```
\displaystyle
g(x) = x^2
```

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Q53b

Find expressions for f(g(x)) and g(f(x)), and state their domains.

```
\displaystyle
f(x)= |x|
```

, ```
\displaystyle
g(x) = x^2 -6
```

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Q53c

Find expressions for f(g(x)) and g(f(x)), and state their domains.

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

, ```
\displaystyle
g(x) = 3x + 2
```

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Q53d

Find expressions for f(g(x)) and g(f(x)), and state their domains.

```
\displaystyle
f(x)= (x + 3)^2
```

, ```
\displaystyle
g(x) = \sqrt{x -3}
```

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Q53e

Find expressions for f(g(x)) and g(f(x)), and state their domains.

```
\displaystyle
f(x)= \log x
```

, ```
\displaystyle
g(x) =3^{x + 1}
```

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Q53f

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

and `g(x) =\sqrt{x}`

.

a) Determine `f(g(x))`

b) State the domain of `f(g(x))`

c) Determine whether `f(g(x))`

is even, odd, or neither.

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Q54

Verify, algebraically, that `f(f^{-1}(x)) = x`

for each of the following.

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

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Q55a

Verify, algebraically, that `f(f^{-1}(x)) = x`

for each of the following.

```
\displaystyle
f(x) = \sin x
```

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Q55b

Verify, algebraically, that `f(f^{-1}(x)) = x`

for each of the following.

```
\displaystyle
f(x) = 3x
```

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Q55c

Verify, algebraically, that `f(f^{-1}(x)) = x`

for each of the following.

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

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Q55d

A Ferris wheel rotates such that the angle of rotation , `\theta`

, is defined by `\theta = \frac{\pi t}{15}`

, where `t`

is the time, in seconds. A rider’s height, `h`

, in metres,
above the ground can be modelled by the function `h(\theta) = 20 \sin \theta + 22`

.

a) Write an equation for the rider’s height in terms of time.

b) Sketch graphs of the three functions, on separate sets of axes, one above the other.

c) Compare the periods of the graphs of `h(\theta)`

and `h(t)`

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