Cumulative Functions Review
Functions 11 McGraw-Hill
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State the domain and the range of each relation. Is each relation a function? Justify your answer.
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\displaystyle \{(-2, 1), (-1, 4), (0, 9), (1, 16), (2, 25) \}
\displaystyle y = 0.5x^2 -4
Write each function in mapping notation. Then, for each function, determine f(-1).
a) f(x) = \sqrt{1 -3x}
b) \displaystyle f(x) = \frac{2x + 1}{x^2 -4}
The amount, A, in dollars, to be invested at an interest rate i to have $1500 after 1 year is given by the relation A(i) = \frac{1500}{1 +i},
Note that i must be expressed as a decimal.
a) Determine the domain and the range for this relation.
b) Graph the relation.
c) How much money needs to be invested at 3%?
d) What rate of interest is required if $1000 is invested?
a) Draw the mapping diagram for the given data.
\displaystyle \{(2, 4), (5, 0), (6, 4), (3, 3), (4, -2)\}
b) Is this relation a function? Explain.
A farmer has 4000 m of fencing to enclose a rectangular field and subdivide it into three equal plots of land. Determine the dimensions of each plot of land so that the total area is a maximum.
A store sells T-shirts with logos on them. Last year, the store sold 600 of these T-shm's at $15 each. The sales manager is plannm'g to increase the price. A survey indicates that for each $1 increase in the price, 30 fewer T- shirts will be sold per year.
a) What price will maximize the yearly revenue?
b) What is the maximum yearly revenue?
Solve each quadratic equation. Give exact answers.
\displaystyle 2x^2 -4x -3 = 0
\displaystyle 3x^2 -12 x + 4 = 0
Use the discriminant to determine the number of roots for each equation.
\displaystyle 4x^2 + 3x - 2 = 0
\displaystyle -3x^2 + 10x - 7 = 0
\displaystyle 5x^2 -8x + 1 = 0
The length of a rectangle is 2 m more than three times its width. If the area is 20 m^2, find the dimensions of the rectangle to the nearest hundredth of a metre.
Determine the equation in standard form for each quadratic function.
x-intercepts -3 and 4, containing the point (1, -4)
x-intercepts 2 \pm \sqrt{3}, y-intercept 2
Water shoots out of a decorative fountain, making an arc in the shape of a parabola. The arc spans a distance of 6 m from one side of the fountain to the other. The height of the arc at a horizontal distance of 1 m from the starting point is 5 m.
a) Sketch the quadratic function that represents the arc such that the vertex of the parabola is on the y—axis and the horizontal distance from one side to the other is along the X-axis.
b) Determine the equation for the function.
c) Find the maximum height of the arc.
Determine the exact point(s) of intersection of each pair of functions.
\displaystyle f(x) = 2x^2 -3x +4 and \displaystyle g(x) =2x + 5
\displaystyle f(x) = -x^2 + 8x + 3 and \displaystyle g(x) = -0.5x + 1
For what value of k will the line y = -5x + k be tangent to the graph of the function \displaystyle f(x) = -4x^2 +3x + 1 ?
Determine whether the functions in each pair are equivalent.
\displaystyle f(x) = (2x + 1)(x- 3) - (x + 2)(x -4) , \displaystyle g(x) = 3(x-1)^2 - (2x + 1)(x-2)
\displaystyle f(x) = (x + 3)(x-2) + 2(x -5)(x + 1) , \displaystyle g(x) = -(x -2)(x + 3) + 2x (1-x)
Given the graph of a function f(x), sketch the graph of g[x) by determining the image points A', B', C', D', and E'
g(x) = f(x) + 4
g(x) = f(x -2)
g(x) = f(x -6) + 3
Q15c
g(x) = f(x + 5) - 1
Q15d
Copy the graph of f(x) and sketch each reflection, g(x). Then, state the domain and range of each function.
g(x) = f(-x)
g(x) = -f(x)
g(x) = -f(-x)
Sketch g(x). Show the transformation mapping.
\displaystyle g(x) =(x + 2)^2 -1
\displaystyle g(x) = \sqrt{x + 3} -4
\displaystyle g(x) = \frac{1}{x -4} + 6
\displaystyle g(x) = (x - 7)^2 + 3
David and his friend Shane are in a 20—km bike race. David bikes 1.5 km/h faster than Shane. The time, in hours, to complete the race is given by t =\frac{d}{v}, where d is the distance, in kilometres, and V is the speed, in kilometres per hour.
a) If V represents Shane's speed, determine a function to represent David's time. What are the domain and range?
b) Determine a function to represent Shane's time. What are the domain and range?
c) Graph both functions on the same set of axes.
d) Use the graph to determine what would be true about Shane's time if it took David 45 min to complete the race.
Determine the equation of each function after
i) a reflection in the x-axis, giving g(x).
ii) a reflection in the y-axis, giving h(x).
f(x) = 2x^2 - 7x +3
f(x) = \sqrt{x} - 3
f(x) = \frac{1}{x + 2}
Given the function f(x) = x^2, identify the value of a or k, transform the graph of f(x) to sketch the graph of g(x), and state the domain and range of g(x).
g(x) = 2f(x)
g(x) = f(3x)
g(x) = f(\frac{x}{4})
g(x) = \frac{1}{3}f(x)
Sketch the function. Show your transformation mapping.
\displaystyle g(x) = 7x
\displaystyle g(x) = \frac{1}{5x}
\displaystyle g(x) = (3x)^2
\displaystyle g(x) = \sqrt{6x}
g(x) = 4f(x+ 3) when f(x) = \sqrt{x}
g(x) = -f(5x) when f(x) = x
g(x) = -3f(2x + 9) - 4 when f(x) = x^2
For the function f(x)
i) determine f^{-1}(x)
ii) graph f(x) and its inverse
iii) determine whether f^{-1}(x) is a function.
f(x) = 11 x - 3
f(x) = 3x^2 +4
f(x) = (x + 8)^2 + 19
f(x) = (x+ 8)^2 +19
Issa works at a furniture store. She earns $450 per week, plus commission of 6% of her sales.
a) Write a function to describe Issa's total weekly earnings as a function of her sales.
b) Determine the inverse of this function.
c) What does the inverse represent?
d) One week, 1553 earned $1020. Calculate her sales that week.
Match each graph with one of these equations.
y =x^2, y = 3x, y = 3^x, y = (\frac{1}{3})^x
A bacterial colony with an initial population of 150 triples every day. Write an equation that models this exponential growth.
A particular radioactive substance has a half-life of 3 years. Suppose an initial sample has a mass of 200 mg.
a) Write the equation that relates the mass of radioactive material remaining to time.
b) How much will remain after one decade?
c) How long will it take for the sample to decay to 10% of its initial mass? Explain how you arrived at your answer.
d) Show how you can write the equation from part a) in another way.
e) Explain why the two equations are equivalent.
Evaluate. Express as a fraction in lowest terms.
\displaystyle 9^{-1}
\displaystyle 4^{-2} + 16^{-1}
\displaystyle 3^{-3} + 3^{0}
\displaystyle (- \frac{1}{5})^{-2}
Q28e
\displaystyle (\frac{2}{3})^{-5}
Q28f
Simplify. Express your answers using only positive exponents.
\displaystyle (x^{-3})(x^{-2})(x^0)
\displaystyle (2nm^2)^{-3}(4n^{-2}m^{-2})
\displaystyle a{-4} \div a^{-5}
\displaystyle \frac{m^{-3}n^{-4}}{m^{-2}b^{-1}}
\displaystyle (x^{-4})^{-5}
\displaystyle (3ab^{-3})^{-2}
Evaluate. Express any fractions in lowest terms.
\displaystyle \sqrt[4]{81}
\displaystyle \sqrt[3]{-1000}
\displaystyle \sqrt[9]{-512}
\displaystyle 343^{\frac{1}{3}}
\displaystyle (\frac{125}{216})^{\frac{1}{3}}
\displaystyle (81)^{\frac{3}{4}}
Q30g
\displaystyle -5^{-4}
Q30h
Q30i
a) Graph the function \displaystyle y = 64(\frac{1}{4})^x
b) State
domain, range
x and y intercepts, if they exist
intervals of increase/decrease
equation of the asymptote
A radioactive substance with an initial mass of 100 mg has a half-life of 1.5 days.
a) Write an equation to relate the mass remaining to time.
b) Graph the function. Describe the shape of the curve.
c) Limit the domain so that the model accurately describes the situation.
d) Find the amount remaining after
i) 8 days ii) 2 weeks
e) How long will it take for the sample to decay to 3% of its initial mass?