1. Introduction to Functions
In everyday life, many quantities depend on one or more changing variables eg:
(a) plant growth depends on sunlight and rainfall
(b) speed depends on distance travelled and time taken
(c) voltage depends on current and resistance
(d) test marks depend on attitude, listening in lectures and doing tutorials (among many other variables!!)
FUNCTIONS
A function is a rule that relates how one quantity depends on other quantities. For example,
(a) V = IR where
V = voltage (V)
I = current (A)
R = resistance (Ω)
If I increases, so does the voltage (assuming resistance is constant).
If R increases, so does the voltage (assuming current is constant).
(b) 
where
s = speed (m / s)
d = distance (m)
t = time taken (s)
If d increases, the speed goes up (assuming time is constant).
If t increases, the speed goes down (assuming distance is constant).
Definition of a Function
Whenever a relationship exists between two variables (or quantities) such that for every value of the first, there is only one corresponding value of the second, then we say:
the second variable is a function of the first variable.
The first variable is the independent variable (usually x), and the second variable is the dependent variable (usually y).
The independent variable and the dependent variable are real numbers.
Example 1:
We know the equation for the area of a circle from primary school:
A = πr2
This is a function as each value of the independent variable r gives us one value of the dependent variable A.
Here is a LiveMath document to illustrate this function.
General Cases
We use x (independent) and y (dependent)
variables for general cases.
Example 2:
In the equation
y = 3x + 1,
y is a function of x, since for each value of x, there is only one value of y.
If we substitute x = 5, we get y = 16 and no other value.
The values of y we get depend on the values chosen for x.
Therefore, x is the independent variable and y is the dependent variable.
Example 3
The force F required to accelerate an object of mass 5 kg by an acceleration of a ms-2 is given by: F = 5a.
Here, F is a function of the acceleration, a.
The dependent variable is F and the independent variable is a.
Function Notation
We normally write functions as: f(x) and read this as "function f of x".
We can use other letters for functions. Common ones are g(x) and h(x). But there are also ones like P(t) which could indicate power at time t.
Example 4
We often come across functions like: y = 2x2+ 5x + 3
We can write this using function notation:
f(x) = 2x2 + 5x + 3
Function notation is all about substitution.
The value of the function f(x) when x = a is written as f(a).
Example 5
If we have f(x) = 4x + 10, the value of f(x) for x = 3 is written:
f(3) = 4 × 3 + 10 = 22
When x = 3, the value of the function f(x) is 22.
Mathematical Notation
Mathematics is often confusing because of the way it is written.
We write 5(102) and it means 5 × 102 = 500.
But if we write a(102), this could mean
- "function a of 102" (that is, the value of the function a when the independent variable is 102) or it could mean
- a × 102 = 100a.
You have to be careful with this.
Also, be careful when substituting letters or expressions into functions.
See a discussion on this: Towards more meaningful math notation.
Example 6
If h(x) = dx3 + 5x then value of h(x) for x = 10 is:
h(10) = d(10)3 + 5(10)
= 1000d + 50
Here is a LiveMath document to illustrate functions.
Example 7
If the height of an object at time t is given by
h(t) = 10t2 − 2t, then
a. The height at time t = 4 is
h(4) = 10(4)2 − 2(4) = 10 ×16 − 8 = 152
b. The height at time t = b is
h(b) = 10b2 − 2b
c. The height at time t = 3b is
h(3b) = 10(3b)2 − 2(3b) = 10 × 9b2 − 6b = 90b2 − 6b
d. The height at time t = b + 1 is
h(b + 1) = 10(b + 1)2 − 2(b + 1) = 10 × (b2 + 2b + 1) − 2b − 2
= 10b2 + 20b + 10 − 2b − 2
= 10b2 + 18b + 8
Exercises:
Evaluate the following functions:
(1) Given f(x) = 3x + 20, find
a. f(-4) b. f(10)
(2) Given that the height of a particular object at time t is
h(t) = 50t − 4.9t2, find
a. h(2) b. h(5)
(3) The voltage, V, in a circuit is a function of time t, and is given by:
V(t) = 3t − 1.02t
Find the voltage at time
a. t = 4 b. t = c + 10
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