Functions and graphs

Exercise 1

For each of the following functions, write down the asymptotes and the domain and range.

a

$f (x) = 4 - \frac{4}{x}$

b

$g (x) = \frac{4 - x}{x}$

c

$h (x) = \frac{x}{x^{2} - 4}$

d

$k (x) = \frac{x^{2}}{x^{2} + 4}$

Exercise 2

The pitch of a sound is determined by its frequency. The higher the frequency, the shorter the wavelength. Frequency is measured in Hertz (Hz) and is equivalent to the number of oscillations per second. If you know the frequency of a wave $f$ then you can calculate its wavelength $W$ with the follwing formula:
$W = \frac{330}{f}$
A sound system can produce sounds in the frequency range from $15$ Hz to $30000$ Hz.

a

If you choose $[15; 30000]$ as the domain, then what is the range of $W$ ?

b

Bats can hear high-frequency sounds, in some cases up to a frequency of $120000$ Hz. Are these high-pitched or low-pitched sounds? What is the corresponding wavelength?

c

Humans are barely able to hear sounds with a frequency lower than $20$ Hz. Are those low-pitched or high-pitched sounds? What is the wavelength of such a sound?

d

What value does $W$ approach as $f$ gets larger?

Exercise 3

You are given the function $f$ with $f (x) = \frac{10 x}{{(x - 20)}^{2}}$ .

a

What is the `x` -intersect of this function?

b

What are the asymptotes of this function?

c

What window settings do you need to use in order to get a proper view of the graph of $f$ with all its characteristics?

d

Determine the range of `f` . (round to two decimals)

Exercise 4

The total costs ( $T C$ ) for the production of an item are given by:
$T C = 100 + 0.1 q^{2}$ , where $q$ is the number of items produced.

a

Calculate the average costs per item given a production of $120$ pieces, rounded to two decimals.

b

Explain why the average costs are equivalent to the slope of the line through points $(0, 0)$ and $(q, T C)$ .

c

Draw up a function rule for the average costs per item ( `AC` ) as a function of $q$ .

d

What is the asymptote of the function `AC` ? Write down the domain and range of `AC` .

Exercises