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

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

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

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

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

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.

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.

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.

Determine the equation of the function.

Determine the equation of the function.

Given the function f(x) = -2x^2 + 1, describe the slope and the change in slope for the appropriate intervals.

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)

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)

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)

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

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

Factor if possible.

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

Factor if possible.

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

Factor if possible.

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

Factor if possible.

\displaystyle x^4 + x^2 + 1

Use the remainder theorem to determine the remainder.

4x^3 -7x^2 + 3x + 5 divided by x -5

Use the remainder theorem to determine the remainder.

6x^4 +7x^2 - 2x -4 divided by 3x +2

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

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

Solve for x.

\displaystyle x^4-81 = 0

Solve for x.

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

Solve.

\displaystyle 8x^3 + 27 = 0

Solve.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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}

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}

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}

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}

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}

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}

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.

Solve algebraically.

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

Solve algebraically.

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

Solve algebraically.

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

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

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

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

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

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.

Determine the exact radian measure for each angle.

\displaystyle 135^o

Determine the exact radian measure for each angle.

\displaystyle -60^o

Determine the exact degree measure for each angle.

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

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

A sector angle of a circle with radius 9 cm measures \displaystyle \frac{5\pi}{12} . What is the perimeter of the sector?

Determine the exact value of each trigonometric ratio.

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

Determine the exact value of each trigonometric ratio.

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

Determine the exact value of each trigonometric ratio.

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

Determine the exact value of each trigonometric ratio.

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

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

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

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

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

Prove each identity.

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

Prove each identity.

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

Prove each identity.

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

Prove each identity.

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

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).

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.

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).

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

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

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})]

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

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

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

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

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

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

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

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

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.

Express in logarithmic form.

a) \displaystyle 7^2 =49

b) \displaystyle a^b = c

c) \displaystyle 8^3 = 512

d) \displaystyle 11^x = y

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.

Express in exponential form.

a) \displaystyle \log_36561 = 8

b) \displaystyle \log_a75 = b

c) \displaystyle \log_72401 = 4

d) \displaystyle \log_a19 = b

Evaluate.

\displaystyle \log_2256

Evaluate.

\displaystyle \log_{15} 15

Evaluate.

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

Evaluate.

\displaystyle \log_{3} 243

Evaluate.

\displaystyle \log_{12} 12

Evaluate.

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

Solve for x.

\displaystyle \log_3x =4

Solve for x.

\displaystyle \log_x125 =3

Solve for x.

\displaystyle \log_7x =5

Solve for x.

\displaystyle \log_x729 =6

Solve for x.

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

Solve for x.

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

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?

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?

Solve each equation. Check for extraneous roots.

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

Solve each equation. Check for extraneous roots.

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

Use the laws of logarithms to evaluate.

\displaystyle \log_84 + \log_8128

Use the laws of logarithms to evaluate.

\displaystyle \log_7 7\sqrt{7}

Use the laws of logarithms to evaluate.

\displaystyle \log_5 10 -\log_5250

Use the laws of logarithms to evaluate.

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

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

Solve.

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

Solve.

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

Determine the point(s) of intersection of the functions f(x) = \log x and g(x) = \frac{1}{2}\log(x + 1).

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.

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?

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%.

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)

Given f(x) = 2x^2 + 3x - 5 and g(x) =x + 3, determine

\displaystyle f(g(x))

Given f(x) = 2x^2 + 3x - 5 and g(x) =x + 3, determine

\displaystyle g(f(x))

Given f(x) = 2x^2 + 3x - 5 and g(x) =x + 3, determine

\displaystyle f(g(-3))

Given f(x) = 2x^2 + 3x - 5 and g(x) =x + 3, determine

\displaystyle g(f(7))

If f(x) = \frac{1}{x} and g(x) = 4 -x, determine the following

\displaystyle f(g(3))

If f(x) = \frac{1}{x} and g(x) = 4 -x, determine the following

\displaystyle f(g(0))

If f(x) = \frac{1}{x} and g(x) = 4 -x, determine the following

\displaystyle f(g(4))

If f(x) = \frac{1}{x} and g(x) = 4 -x, determine the following

\displaystyle g(f(4))

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

\displaystyle f(x)= \sqrt{x} , \displaystyle g(x) = x + 1

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

\displaystyle f(x)= \sin x , \displaystyle g(x) = x^2

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

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

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

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}

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}

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.

Verify, algebraically, that f(f^{-1}(x)) = x for each of the following.

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

Verify, algebraically, that f(f^{-1}(x)) = x for each of the following.

\displaystyle f(x) = \sin x

Verify, algebraically, that f(f^{-1}(x)) = x for each of the following.

\displaystyle f(x) = 3x

Verify, algebraically, that f(f^{-1}(x)) = x for each of the following.

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

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).