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Solutions
45 Videos

Determine whether the functions in pair are equivalent.

```
\displaystyle
f(x) = (x +6)(x - 8) + (x + 16)(x + 3)
```

```
\displaystyle
g(x) = 3(x^2 + 3x + 5) -(x -5)(x - 3)
```

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Q1a

Determine whether the functions in pair are equivalent.

```
\displaystyle
f(x) = (x + 5)(x -4) -(x -8)(x -1)
```

```
\displaystyle
g(x) = 2(5x - 28)
```

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Q1b

Simplify each expression and state all
restrictions on `x`

.

```
\displaystyle
\frac{x + 7}{x^2 + 10x + 21}
```

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Q2a

Simplify each expression and state all
restrictions on `x`

.

```
\displaystyle
\frac{x^2 - 64}{x -8}
```

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Q2b

A square piece of cardboard with side length 40 cm is used to create an open-topped box by cutting out squares with side length `x`

from each corner.

a) Determine a simplified expression for the surface area of the box.

b) Determine any restrictions on the value of x.

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Q3

Simplify the expression and state the restrictions.

```
\displaystyle
\frac{3x^2 }{5xy} \times \frac{20xy^3}{12 xy}
```

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Q4a

Simplify the expression and state the restrictions.

```
\displaystyle
\frac{15a^3b^4}{20a^2b} \div \frac{6b}{8ab^2}
```

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Q4b

Simplify the expression and state the restrictions.

```
\displaystyle
\frac{1}{3x} + \frac{5}{2x^2}
```

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Q4c

Simplify the expression and state the restrictions.

```
\displaystyle
\frac{4}{x-6} - \frac{3}{x -4}
```

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Q4d

Simplify the expression and state the restrictions.

```
\displaystyle
\frac{x^2 + 7x}{3x + 21} \times \frac{x^2 + 3x + 2}{x + 2}
```

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Q5a

Simplify the expression and state the restrictions.

```
\displaystyle
\frac{x^2 + 4x -60}{3x + 30} \div \frac{x^2 -8x + 12}{6x -12}
```

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Q5b

Simplify the expression and state the restrictions.

```
\displaystyle
\frac{3}{x^2 + 7x +10} - \frac{5x}{x^2 -4}
```

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Q5c

Simplify the expression and state the restrictions.

```
\displaystyle
\frac{-10x}{x^2 +18x + 32} + \frac{12x}{x^2 + 6x -160}
```

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Q5d

A square piece of cardboard with side length 40 cm is used to create an open-topped box by cutting out squares with side length `x`

from each corner.

Determine a simplified expression for the ratio of the volume to the surface area. What are the restrictions on `x`

?

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Q6

Copy the graph of the function `f(x)`

. Sketch the graph of `g(x)`

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

`g(x) = f(x) +6`

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Q7a

Copy the graph of the function `f(x)`

. Sketch the graph of `g(x)`

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

`g(x) = f(x-3) `

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Q7b

Let `f(x) = x^2`

.

- Express
`g(x) = f(x -d) + c`

. - Sketch g(x).
- State the Domain and Range.

`g(x) = (x + 7)^2 -8`

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Q8a

Let `f(x) = x^2`

.

- Express
`g(x) = f(x -d) + c`

. - Sketch g(x).
- State the Domain and Range.

`g(x) = \sqrt{x -6} + 3`

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Q8b

Let `f(x) = x^2`

.

- Express
`g(x) = f(x -d) + c`

. - Sketch g(x).
- State the Domain and Range.

`g(x) = \frac{1}{x + 3}+ 1`

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Q8c

Copy the graph of f(x) and sketch g(x)). State the domain and range of each function.

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

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Q9a

Copy the graph of f(x) and sketch g(x)). State the domain and range of each function.

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

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Q9b

Copy the graph of f(x) and sketch g(x)). State the domain and range of each function.

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

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Q9c

Determine the equation of each function, g(x), after a reflection in the x-axis.

- i)
`f(x) = \sqrt{x} + 5`

- ii)
`f(x) =\frac{1}{x} - 7`

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Q10a

Determine the equation of each function
`h(x)`

, after a reflection in the y—axis.

- i)
`f(x) = \sqrt{x} + 5`

- ii)
`f(x) =\frac{1}{x} - 7`

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Q10b

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 the function.

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

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Q11a

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 the function.

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

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Q11b

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 the function.

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

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Q11c

`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 the function.

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

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Q11d

For each function g(x), describe the transformation from a base function of
`f(x) =x, f(x) = x^2 , f(x) = \sqrt{x}`

, or `f(x) =\frac{1}{x}`

.

Then, transform the graph of `f(x)`

to sketch the graph of `g(x)`

.

`g(x) =5x`

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Q12a

For each function g(x), describe the transformation from a base function of
`f(x) =x, f(x) = x^2 , f(x) = \sqrt{x}`

, or `f(x) =\frac{1}{x}`

.

Then, transform the graph of `f(x)`

to sketch the graph of `g(x)`

.

`g(x) = \frac{1}{4x}`

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Q12b

For each function g(x), describe the transformation from a base function of
`f(x) =x, f(x) = x^2 , f(x) = \sqrt{x}`

, or `f(x) =\frac{1}{x}`

.

Then, transform the graph of `f(x)`

to sketch the graph of `g(x)`

.

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

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Q12c

`f(x) =x, f(x) = x^2 , f(x) = \sqrt{x}`

, or `f(x) =\frac{1}{x}`

.

Then, transform the graph of `f(x)`

to sketch the graph of `g(x)`

.

`g(x) = \sqrt{9x}`

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Q12d

Describe, in the appropriate order, the transformations that must be applied to the base function `f(x)`

to obtain the transformed function. Then, write the corresponding equation and transform the graph of `f(x)`

to sketch the graph of `g(x)`

.

`f(x) =\sqrt{x}, g(x) = 3f(x+ 6)`

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Q13a

Describe, in the appropriate order, the transformations that must be applied to the base function `f(x)`

to obtain the transformed function. Then, write the corresponding equation and transform the graph of `f(x)`

to sketch the graph of `g(x)`

.

`f(x) =x, g(x) = -f(6x)-5`

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Q13b

Describe, in the appropriate order, the transformations that must be applied to the base function `f(x)`

to obtain the transformed function. Then, write the corresponding equation and transform the graph of `f(x)`

to sketch the graph of `g(x)`

.

`f(x)= \frac{1}{x}, g(x) = \frac{1}{5}f(x) + 4`

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Q13c

`f(x)`

to obtain the transformed function. Then, write the corresponding equation and transform the graph of `f(x)`

to sketch the graph of `g(x)`

.

`f(x)=x^2, g(x) = -2f(3x + 12) - 6`

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Q13d

`f(x) =x, f(x) = x^2 , f(x) = \sqrt{x}`

, or `f(x) =\frac{1}{x}`

.

Then, transform the graph of `f(x)`

to sketch the graph of `g(x)`

.

`g(x) = 2x + 9`

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Q14a

`f(x) =x, f(x) = x^2 , f(x) = \sqrt{x}`

, or `f(x) =\frac{1}{x}`

.

Then, transform the graph of `f(x)`

to sketch the graph of `g(x)`

.

`g(x) = \frac{3}{x + 4}`

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Q14b

`f(x) =x, f(x) = x^2 , f(x) = \sqrt{x}`

, or `f(x) =\frac{1}{x}`

.

Then, transform the graph of `f(x)`

to sketch the graph of `g(x)`

.

`g(x) = -4\sqrt{x} = 1`

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Q14c

`f(x) =x, f(x) = x^2 , f(x) = \sqrt{x}`

, or `f(x) =\frac{1}{x}`

.

Then, transform the graph of `f(x)`

to sketch the graph of `g(x)`

.

`g(x) =(5x + 20)^2`

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Q14d

For each function `f(x)`

:

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

- ii) graph
`f(x)`

and its inverse. - ii) state whether or not
`f^{-1}(x)`

is a function.

`f(x)= 7x -5`

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Q15a

For each function `f(x)`

:

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

- ii) graph
`f(x)`

and its inverse. - ii) state whether or not
`f^{-1}(x)`

is a function.

```
\displaystyle
f(x) = 2x^2 + 9
```

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Q15b

For each function `f(x)`

:

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

- ii) graph
`f(x)`

and its inverse. - ii) state whether or not
`f^{-1}(x)`

is a function.

```
\displaystyle
f(x) = (x+ 4)^2 + 15
```

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Q15c

For each function `f(x)`

:

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

- ii) graph
`f(x)`

and its inverse. - ii) state whether or not
`f^{-1}(x)`

is a function.

```
\displaystyle
f(x) = 5x^2 + 20x -10
```

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Q15d

Iai works at an electronics store. She earns $600 a week, plus commission of 5% of her sales.

a) Write a function to describe Jai’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, Iai earned $775. Calculate her sales that week.

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Q16