Functions and graphs

Exercise 1

For the following functions, determine domain and range. Round to two decimals where necessary.

a

$f (x) = x^{2} - x - 6$

b

$g (x) = x^{2} (x - 2) (x - 3)$

c

$h (x) = x^{3} - 6 x$

d

$k (x) = 1 + 2 \sqrt{x}$

Exercise 2

On the right you see the graphs of two functions $f$ and $g$ with $f (x) = x^{2} - 2 x^{4}$ and $g (x) = - x^{2}$ , using the standard window settings.

a

Algebraically determine the zeros of $f$ .

b

The standard window settings are not that useful if you want to see the `x` -intersects and vertices of both functions. Choose better window settings to determine the vertices of $f$ .

c

Determine the range of both functions.

d

Determine the intersection points of the two graphs of $f$ and $g$ algebraically .

Exercise 3

A firework is launched from ground level. Its height above ground level is subsequently dependent on time, until it explodes. The following function describes the trajectory: $h (t) = 40 t - 5 t^{2}$ . In this formula $h$ is the height above ground level in meters, and $t$ is the time in seconds.

a

The firework explodes after $6$ seconds. What is the maximum height it can reach?

b

Write down the domain and range of this function, taking the context into account.

c

At what height did the firework explode?

d

How long (in seconds) would the firework be visible above a row of trees with a height of $40$ m?

e

Why does the graph above not show the trajectory of the firework?

Exercise 4

Every week, a trader can sell up to $400$ pieces of a certain product. He has no competition, and therefore the number $q$ of items he sells is only dependent on the price $p$ he chooses. He finds the following relationship: $q = 400 - 0.5 p$ .

a

Write down a formula of the revenue $R$ as a function of price $p$ .

b

What values are possible for $p$ ?

c

What values are possible for $R$ ?

Exercise 5

Given the function $f$ with $f (x) = -2 {(x - 10)}^{2} + 60$ and domain $[0, 40]$ .

Determine the range of $f$ .

Exercises