Functions and graphs

Exercise 1

Solve the following inequalities algebraically.

a

$x^{3} > x$

b

$x^{3} \leq 80 x - 2 x^{2}$

c

$\frac{8}{x^{2}} \geq x$

d

$x^{2} - 4 x > -3$

Exercise 2

You can drive a Smart Fortwo for as little as € 5.00 per day! Imagine that you had bought such a car on January $1$ , 2006 and paid that $5$ euro per day. To cover the maintenance costs they offer you a subscription of $1.5$ cent per kilometre driven, which will cover just about all maintenance fees. You are then left with just the fuel costs. With $1$ litre of fuel you can drive $15$ kilometres. The price of $1$ litre of fuel is approximately € 1.50.

a

How much do you pay per kilometre for petrol and maintenance combined?

b

What would this Smart cost you if you drove $16000$ km per year?

c

Draw up the inequality for the following question: What is the maximum distance (in kilometres) that you could drive this Smart if you wanted to spend less than € 4000 that year? Then proceed to solve this inequality algebraically.

d

The maintenance subscription of $1.5$ cent per kilometre is only valid if you drive at least $15000$ km/year. If your distance travelled is lower, then you will be charged for $15000$ km/year. Write down the complete function rule for the annual costs $K$ as a function of distance travelled.

Exercise 3

Two cars are travelling along the highway M1, both with (approximately) constant speed. Driver A maintains a speed of $110$ km/h. Driver B has a speed of $120$ km/h. When driver B crosses the bridge across the river IJssel at the city of Deventer he is $24$ kilometres behind driver A. This happens at time $t = 0$ . The distance (in kilometres) to Deventer is $a (t)$ .

a

For each car, draw up a linear function for $a (t)$ .

b

Calculate how many minutes it takes car B to catch up with car A.

c

Algebraically determine in which time interval the distance between the two cars is less than $4$ kilometers.

Exercise 4

Given function $f$ with $f (x) = (x^{2} - 4) (x^{2} - 9)$ .

a

Solve algebraically: $f (x) \leq 0$ .

b

Solve algebraically: $f (x) < 36$ .

Exercises