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Solutions
79 Videos

State the domain and the range of each relation. Is each relation a function? Justify your answer.

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Q1a

State the domain and the range of each relation. Is each relation a function? Justify your answer.

```
\displaystyle
\{(-2, 1), (-1, 4), (0, 9), (1, 16), (2, 25) \}
```

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Q1b

State the domain and the range of each relation. Is each relation a function? Justify your answer.

```
\displaystyle
y = 0.5x^2 -4
```

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Q1c

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}
```

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Q2

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?

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Q3

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.

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Q4

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.

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Q5

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?

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Q6

Solve each quadratic equation. Give exact answers.

```
\displaystyle
2x^2 -4x -3 = 0
```

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Q7a

Solve each quadratic equation. Give exact answers.

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

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Q7b

Use the discriminant to determine the number of roots for each equation.

```
\displaystyle
4x^2 + 3x - 2 = 0
```

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Q8a

Use the discriminant to determine the number of roots for each equation.

```
\displaystyle
-3x^2 + 10x - 7 = 0
```

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Q8b

Use the discriminant to determine the number of roots for each equation.

```
\displaystyle
5x^2 -8x + 1 = 0
```

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Q8c

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.

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Q9

Determine the equation in standard form for each quadratic function.

x-intercepts -3 and 4, containing the point (1, -4)

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Q10a

Determine the equation in standard form for each quadratic function.

x-intercepts `2 \pm \sqrt{3}`

, y-intercept 2

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Q10b

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.

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Q11

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
```

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Q12a

Determine the exact point(s) of intersection of each pair of functions.

```
\displaystyle
f(x) = -x^2 + 8x + 3
```

and ```
\displaystyle
g(x) = -0.5x + 1
```

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Q12b

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
```

?

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Q13

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)
```

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Q14a

Determine whether the functions in each pair are equivalent.

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

, ```
\displaystyle
g(x) = -(x -2)(x + 3) + 2x (1-x)
```

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Q14b

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`

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Q15a

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 -2) `

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Q15b

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 -6) + 3`

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Q15c

`f(x)`

, sketch the graph of `g[x)`

by determining the image
points A', B', C', D', and E'

`g(x) = f(x + 5) - 1`

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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)`

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Q16a

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)`

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Q16b

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)`

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Q16c

Sketch `g(x)`

. Show the transformation mapping.

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

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Q17a

Sketch `g(x)`

. Show the transformation mapping.

```
\displaystyle
g(x) = \sqrt{x + 3} -4
```

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Q17b

Sketch `g(x)`

. Show the transformation mapping.

```
\displaystyle
g(x) = \frac{1}{x -4} + 6
```

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Q17c

Sketch `g(x)`

. Show the transformation mapping.

```
\displaystyle
g(x) = (x - 7)^2 + 3
```

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Q17d

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.

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Q18

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`

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Q19a

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) = \sqrt{x} - 3`

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Q19b

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) = \frac{1}{x + 2}`

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Q19c

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)`

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Q20a

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) = f(3x)`

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Q20b

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) = f(\frac{x}{4})`

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Q20c

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

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Q20d

Sketch the function. Show your transformation mapping.

```
\displaystyle
g(x) = 7x
```

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Q21a

Sketch the function. Show your transformation mapping.

```
\displaystyle
g(x) = \frac{1}{5x}
```

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Q21b

Sketch the function. Show your transformation mapping.

```
\displaystyle
g(x) = (3x)^2
```

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Q21c

Sketch the function. Show your transformation mapping.

```
\displaystyle
g(x) = \sqrt{6x}
```

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Q21d

Sketch the function. Show your transformation mapping.

`g(x) = 4f(x+ 3)`

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

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Q22a

Sketch the function. Show your transformation mapping.

`g(x) = -f(5x)`

when `f(x) = x`

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Q22b

Sketch the function. Show your transformation mapping.

`g(x) = -3f(2x + 9) - 4`

when `f(x) = x^2`

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Q22c

For the function `f(x)`

determine

- 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`

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Q23a

For the function `f(x)`

determine

- i) determine
`f^{-1}(x)`

- ii) graph f(x) and its inverse
- iii) determine whether
`f^{-1}(x)`

is a function.

`f(x) = 3x^2 +4`

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Q23b

For the function `f(x)`

determine

- i) determine
`f^{-1}(x)`

- ii) graph f(x) and its inverse
- iii) determine whether
`f^{-1}(x)`

is a function.

f(x) = (x + 8)^2 + 19

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Q23c

For the function `f(x)`

determine

- i) determine
`f^{-1}(x)`

- ii) graph f(x) and its inverse
- iii) determine whether
`f^{-1}(x)`

is a function.

`f(x) = (x+ 8)^2 +19`

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Q23d

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.

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Q24

Match each graph with one of these equations.

`y =x^2, y = 3x, y = 3^x, y = (\frac{1}{3})^x`

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Q25

A bacterial colony with an initial population of 150 triples every day. Write an equation that models this exponential growth.

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Q26

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.

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Q27

Evaluate. Express as a fraction in lowest terms.

```
\displaystyle
9^{-1}
```

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Q28a

Evaluate. Express as a fraction in lowest terms.

```
\displaystyle
5^{-2}
```

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Q28b

Evaluate. Express as a fraction in lowest terms.

```
\displaystyle
4^{-2} + 16^{-1}
```

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Q28c

Evaluate. Express as a fraction in lowest terms.

```
\displaystyle
3^{-3} + 3^{0}
```

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Q28d

Evaluate. Express as a fraction in lowest terms.

```
\displaystyle
(- \frac{1}{5})^{-2}
```

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Q28e

Evaluate. Express as a fraction in lowest terms.

```
\displaystyle
(\frac{2}{3})^{-5}
```

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Q28f

Simplify. Express your answers using only positive exponents.

```
\displaystyle
(x^{-3})(x^{-2})(x^0)
```

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Q29a

Simplify. Express your answers using only positive exponents.

```
\displaystyle
(2nm^2)^{-3}(4n^{-2}m^{-2})
```

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Q29b

Simplify. Express your answers using only positive exponents.

```
\displaystyle
a{-4} \div a^{-5}
```

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Q29c

Simplify. Express your answers using only positive exponents.

```
\displaystyle
\frac{m^{-3}n^{-4}}{m^{-2}b^{-1}}
```

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Q29d

Simplify. Express your answers using only positive exponents.

```
\displaystyle
(x^{-4})^{-5}
```

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Q29e

Simplify. Express your answers using only positive exponents.

```
\displaystyle
(3ab^{-3})^{-2}
```

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Q29f

Evaluate. Express any fractions in lowest terms.

```
\displaystyle
\sqrt[4]{81}
```

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Q30a

Evaluate. Express any fractions in lowest terms.

```
\displaystyle
\sqrt[3]{-1000}
```

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Q30b

Evaluate. Express any fractions in lowest terms.

```
\displaystyle
\sqrt[9]{-512}
```

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Q30c

Evaluate. Express any fractions in lowest terms.

```
\displaystyle
343^{\frac{1}{3}}
```

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Q30d

Evaluate. Express any fractions in lowest terms.

```
\displaystyle
(\frac{125}{216})^{\frac{1}{3}}
```

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Q30e

Evaluate. Express any fractions in lowest terms.

```
\displaystyle
(81)^{\frac{3}{4}}
```

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Q30f

Evaluate. Express any fractions in lowest terms.

```
\displaystyle
128^{\frac{4}{7}}
```

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Q30g

Evaluate. Express any fractions in lowest terms.

```
\displaystyle
-5^{-4}
```

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Q30h

Evaluate. Express any fractions in lowest terms.

```
\displaystyle
(\frac{2}{3})^{-5}
```

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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

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Q31

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?

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Q33