# CLASS-11RELATION & 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.