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Solutions
35 Videos

For each relation, determine the domain and range and whether the relation is a function. Explain your reasoning.

**(a)** {`(-3, 0), (-1, -1), (0, 1), (4, 5), (0, 6)`

}

**(b)** `y = 4-x`

**(c)**

**(d)** `x^2 + y^2 = 16 `

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1.30mins

Q1

What rule can you use to determine, from the graph of a relation, whether the relation is a function? Graph the relation and determine which are functions.

{`(-2,1), (1, 1), (0, 0), (1, -1), (1, -2), (2, -2)`

}

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Q2a

What rule can you use to determine, from the graph of a relation, whether the relation is a function? Graph the relation and determine which are functions.

`y = 4- 3x`

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Q2b

What rule can you use to determine, from the graph of a relation, whether the relation is a function? Graph the relation and determine which are functions.

`y = (x - 2)^2+ 4`

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0.33mins

Q2c

`x^2 + y^2 = 1`

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Q2d

`y = \frac{1}{x}`

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Q2e

`y = \sqrt{x}`

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Q2f

Sketch the graph of a function whose domain is the set of real numbers and whose range is the set of real numbers less than or equal to 3.

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0.48mins

Q3

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

and `g(x) = 2x - 3`

, determine each.

```
\displaystyle
\begin{array}{ccccccc}
&a) & f(-1) & d)& f(2b)\\
&b) &f(0) & e)& g(1- 4a)\\
&c)& g(\frac{1}{2}) & f) & x \text{ when } f(x) = g(x) \\
\end{array}
```

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Q4

**(a)** Graph the function `f(x) =-2(x - 3)^2 + 4`

, and state its domain and range.

**(b)** What does `f(1)`

represent on the graph?

**(c)** Use the equation to determine each of the following.

```
\displaystyle
\begin{array}{cccccc}
&(i) &f(3) -f(2) &(iii) & f(1-x) \\
&(ii) &2f(5) + 7 &&\\
\end{array}
```

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1.08mins

Q5ab

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

, determine the input(s) for `x`

whose output is `f(x) = 8`

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0.29mins

Q6

A ball is thrown upward from the roof of a building 60 m tall. The ball reaches a height of 80 m above the ground after 2 s and hits the ground 6 s after being thrown.

**(a)** Sketch a graph that shows the height of the ball as a function of time.

**(b)** State the domain and range of the function.

**(c)** Determine an equation for the function.

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7.07mins

Q7

State the domain and range of each function.

**(a)** `f(x) = 2(x - 1)^2 + 3`

**(b)** `f(x) = \sqrt{2x + 4}`

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1.07mins

Q8

A farm has `540 m`

of fencing to enclose a rectangular area and divide in into two sections as shown.

**(a)** Write an equation to express the total area enclosed as a function to the width.

**(b)** Determine the domain and range of this area function.

**(c)** Determine the dimensions that give the maximum area.

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3.55mins

Q9

Determine the inverse functions for the following algebraically for (b) and graphically for (a).

**(a)** ```
\displaystyle
f(x) = 2x - 5
```

**(b)** ```
\displaystyle
f(x) = 4- \frac{1}{2}x
```

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1.27mins

Q10ab

For a fundraising event, a local charity organization expects to receive `\$15 000`

from corporate sponsorship, plus `\$30`

from each person who attends the event.

**(a)** Use function notation to express the total income from the event as a function of the number of people who attend.

**(b)** Suggest a reasonable domain and range for the function in part (a). Explain your reasoning.

**(c)** The organizers want to know how many tickets they need to sell to reach their fundraising goal. Create a function to express the number of people as a function of expected income. State the domain of this new function.

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Q11

In each graph, a parent function has undergone a transformation of the form `f(kx)`

. Determine the equations of the transformed functions graphed in red. Explain your reasoning.

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0.45mins

Q12a

`f(kx)`

. Determine the equations of the transformed functions graphed in red. Explain your reasoning.

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2.19mins

Q12b

For the set of functions, transform the graph of `f(x)`

to sketch `g(x)`

and `h(x)`

, and state the domain and range of each function.

```
\displaystyle
f(x) = x^2, g(x) = (\frac{1}{2}x)^2, h(x) = -(2x)^2
```

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1.12mins

Q13a

For the set of functions, transform the graph of `f(x)`

to sketch `g(x)`

and `h(x)`

, and state the domain and range of each function.

```
\displaystyle
f(x) =|x|, g(x) = |-4x|, h(x) = |\frac{1}{4}x|
```

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1.15mins

Q13b

Three transformations are applied to `y= x^2`

: a vertical stretch by the factor 2, a translation 3 units right, and a translation 4 units down.

**(a)** Is the order of the transformations important?

**(b)** Is there any other sequence of these transformations that could produce the same result?

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1.39mins

Q14

The point `(1, 4)`

is on the graph of `y=f(x)`

. Determine the coordinates of the image of this point on the graph of `y = 3f[-4(x + 1)] -2`

.

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0.50mins

Q15

State the transformation mapping and describe all the transformation for `y = -2f(\frac{1}{3}x + 4)- 1`

.

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1.38mins

Q16a

Graph `y = -2f(\frac{1}{3}x + 4) - 1`

for `f(x) = x^2`

.

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1.23mins

Q16b

Write the equation for the transformed function, sketch its graph, and state its domain and range.

The graph of `f(x) = \sqrt{x}`

is compressed horizontally by the factor `\frac{1}{2}`

, reflected in the `y-axis`

, and translated `3`

units right and `2`

unit down.

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2.18mins

Q17a

Write the equation for the transformed function, sketch its graph, and state its domain and range.

The graph of `y = \frac{1}{x}`

is stretched vertically by the factor 3, reflected in the x-axis, and translated 34 units left and 1 unit up.

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3.01mins

Q17b

If `f(x) = (x -4)(x + 3)`

, determine the `x-`

intercepts of the function.

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

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Q18a

If `f(x) = (x -4)(x + 3)`

, determine the `x-`

intercepts of the function.

```
\displaystyle
y = -2f(x)
```

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Q18b

If `f(x) = (x -4)(x + 3)`

, determine the `x-`

intercepts of the function.

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

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Q18c

If `f(x) = (x -4)(x + 3)`

, determine the `x-`

intercepts of the function.

```
\displaystyle
y = f(-(x + 1))
```

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Q18d

A function `f(x)`

has domain `\{x \in \mathbb{R} \vert x \geq -4\}`

and range `\{ y\in \mathbb{R}| y < -1\}`

. Determine the domain and range of each function.

`y = 2f(x)`

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Q19a

A function `f(x)`

has domain `\{x \in \mathbb{R} \vert x \geq -4\}`

and range `\{ y\in \mathbb{R}| y < -1\}`

. Determine the domain and range of each function.

`y = 3f(x + 1) + 4`

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1.02mins

Q19b

A function `f(x)`

has domain `\{x \in \mathbb{R} \vert x \geq -4\}`

and range `\{ y\in \mathbb{R}| y < -1\}`

. Determine the domain and range of each function.

`y = f(-x)`

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Q19c

`f(x)`

has domain `\{x \in \mathbb{R} \vert x \geq -4\}`

and range `\{ y\in \mathbb{R}| y < -1\}`

. Determine the domain and range of each function.

`y = -2f(-x + 5) + 1`

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Q19d