What a function is

A function is a rule that gives each number $\displaystyle x$ exactly one number $\displaystyle y$.

• $\displaystyle x$ is the number you choose. It is called the independent variable.
• $\displaystyle y$ is the number the rule gives. It is called the dependent variable, because its value depends on $\displaystyle x$.

The rule “multiply by 2 and add 1” is a function: for each $\displaystyle x$ it gives one $\displaystyle y$.

• $\displaystyle {x}={3}$: $\displaystyle {3\cdot2+1}={7}$, so $\displaystyle {y}={7}$.
• $\displaystyle {x}={0}$: $\displaystyle {0\cdot2+1}={1}$, so $\displaystyle {y}={1}$.
• $\displaystyle {x}={-4}$: $\displaystyle {(-4)\cdot2+1}={-7}$, so $\displaystyle {y}={-7}$.

Each such pair is written as an ordered pair $\displaystyle (x,y)$: $\displaystyle (3,7)$, $\displaystyle (0,1)$, $\displaystyle (-4,-7)$.

A taxi ride costs 12 shekels to start.
Each kilometre costs 3 shekels more.

• $\displaystyle x$ – the length of the ride in kilometres. $\displaystyle y$ – the price in shekels.
• A ride of 5 km: $\displaystyle {12+3\cdot5}={27}$ – the price is 27 shekels.
• Each length of ride has one price, so the price is a function of the length of the ride.

At each hour of the day there is one temperature, so the temperature is a function of the hour.

• $\displaystyle x$ – the hour, $\displaystyle y$ – the temperature.
• At 9 and at 18 the temperature is the same: 18 degrees. A function allows this – two values of $\displaystyle x$ may have the same $\displaystyle y$.

A matching that gives one $\displaystyle x$ two values of $\displaystyle y$ or more is not a function.

• From a temperature to the hour. Turn the table above around: to each temperature match the hour it was measured. The temperature 18 gets two hours, 9 and 18.
• From a number to a number whose square is $\displaystyle x$. 9 gets two numbers, 3 and $\displaystyle -3$, because $\displaystyle {3^2}={9}$ and also $\displaystyle {(-3)^2}={9}$.

The rule: multiply $\displaystyle x$ by 3 and subtract 4.
Find $\displaystyle y$ for each of the values of $\displaystyle x$.

$\displaystyle {x}={5}$

$\displaystyle {x}={0}$

$\displaystyle {x}={-2}$

Is the matching a function? Explain.

1. To each student in the class – their birthday.
2. To each month – the students in the class born in it.
3. To each number – the number 5 greater than it.
4. To each number – any number greater than it.
5. To each number – its opposite.

A phone plan costs 30 shekels a month.
Each minute of a call abroad costs 0.5 shekels more.

• What is the independent variable, and what is the dependent variable?
• How much do you pay in a month in which you talked abroad for 40 minutes?
• One month the bill was 45 shekels. How many minutes did you talk abroad that month? Write an equation and solve it.