# Functions and Limits

In this lecture we shall discuss:- What a function is
- Notation for functions
- Domain of a function
- Range of a function

Function

If a quantity

*y*depends on another quantify

*x*in such a way that each value of

*x*determines exactly

**one**value of

*y*, we say that

*y*is a function of*x***y=4x+1**defines

*y*as a function of

*x*because each value assigned to

*x*determines exactly one value of

*y*.

x |
Value of y = 4x + 1 |

2 | 9 |

1 | 5 |

0 | 1 |

-1/4 | 0 |

√3 | 4√3 + 1 |

In the following example

*y*is not a function of

*x*, as each value assigned to

*x*determines two values of

*y*.

**y = ± √x**

if x = 4

y = ±√4

y = 2 and y = -2

If we use the letter f to denote a function, then the equation

**y = f(x)**

*y*is a function of

*x*

Although

*f*is the symbol most commonly used to denote a function, any symbol can be used. Thus

**y = F(x)**

y = g(x)

y = h(x)

y = g(x)

y = h(x)

Example

φ(x) = 1/(x

^{3}- 1)

Then

φ(

^{3}√7) = 1/(x

^{3}= 1/(

^{3}√7)

^{3}- 1) = 1/(7 - 1) = 1/6

φ(1) = 1/[(1)

^{3}- 1] = 1/0 Undefined

Example

F(x) = 2x

^{2}- 1

F(d) = 2(d)

^{2}- 1

F(t - 1) = 2(t - 1)

^{2}- 1

= 2(t

^{2}-2t + 1) -1

= 2t

^{2}- 4t + 1

g(c) = c

^{2}- 4c

g(x) = x

^{2}- 4x

Range of a Function

For every value given to the independent variable from the domain in a function, we get a corresponding

*y*value.

*The set of all such*

**y**values is called the range of the functionExample

h(x) = 1/[(x - 1)/(x - 3)]

The Domain is

(-∞, 1) ∪ (1, 3) ∪ (3, +∞)

Example

h(x) = (x

^{2}- 4)/(x - 2) = [(x - 2)(x + 2)]/(x - 2) = (x + 2) x ≠ 2

f(x) = x

^{2}

Rewrite as:

y = x

^{2}

Natural Domain

If a function is defined by a formula and there is no domain explicitly stated, then it is understood that the domain consists of all real numbers for which the formula makes sense, and the function has a real value. This is called the natural domain of the function.

Example

y = (x + 1)/(x - 1) - The natural domain is all reals except

**1**

Solve for

*y*

x = (y + 1)/(y - 1) - Range is all reals except

**1**

Piecewise-defined Functions