LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

RELATION & FUNCTION - TERMS VALUE & IMAGE OF A FUNCTION

__The Terms Value and Image of a Function –__

**If, f is a function and if ‘x’
is any element of the domain, we use the symbol f(x) to denote the object which
f associates with ‘x’. f(x) is the functional value at ‘x’. The symbol f(x) is
read as “the value of ‘f ’ at x”, or “f
at x”, or “f of x”. f(x) is called the
image of x and x is called the pre-image of f(x). We could also designate the
function as f : A → B or f : x → f(x)
[read “f takes x into f(x)”]**

**For Example –**

**(i) Let f be {(0, 1), (1, 2),
(2, 3)}. It is a function whose domain is {0, 1, 2} and range is {1, 2, 3}.
Here f(0) = 1, f(1) = 2 and f(2) = 3**

**Euler invented a symbolic way to
write f : → f(x) as y = f(x) while is read “y equals f of x”**

**(ii) Consider the two sets for
the functions f(x) = x²**

** 3 9**

**
A = {-2, 2, ------, 0}, B = {4, ------, 0} **

** 4 16**

** 3 9**

**The set {-2, 2, ------, 0} is
the domain and the set {4, ------, 0}
is **

** 4 16**

**the range, 4 is the image of**

** 9 3**

**each of -2 and 2, ------ is
the image of ----- and 0 is the
image of 0. **

** 16 4**

**If we denote the function by f, then –**

** 3 9**

**f(-2) = 4, f(2) = 4, f(-----)
= ------, f(0) = 0**

** 4 16**

**If, f(x) is a given function of x and if ‘a’ is
in its domain then by f(a) is meant the number obtained by replacing x by ‘a’
in f(x) or the value assumed by f(x) when x = a.**

**Illustration => If f(x) = x²- 5x + 3, then f(1) = (1)²- 5(1)
+ 3 = -1**

** f(-3) = (-3)²- 5(-3) + 3, then
f(1) **

** = 9 + 15 + 3 = 27;**

** f(a) = a²- 5a +3**

**The domain of a function may be
restricted by context. For example, the domain of the area function given by A
= πr² only allows the radius r to be positive. When we define a function y =
f(x) with a formula and the domain is not stated explicitly or restricted by
context, the domain is assumed to be the largest set of real x-values for which
the formula gives real y-values, the so-called natural domain. If we want to
restrict the domain in someway, we must say so.
The domain of y = x² is the entire set of real numbers. To restrict the
function to, say, positive values of x, we would write “y = x², x > 0”**

**Changing the domain to which we
apply a formula usually changes the range as well. The range of y = x² is (0,
∞). The range of y = x², x ≥ 2, is the set of all numbers obtained by
squaring numbers greater than or equal to 2. In set notation, the range is {x²
ǀ x ≥ 2} or {y ǀ y ≥ 4} or [4, ∞).**

**When the range of a function is a set of real
numbers, the function is said to be real valued. The domains and
ranges of many real valued functions of a real variable are intervals or
combinations of intervals. The intervals may be open, closed or half open, and
may be finite or infinite. **