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| Real Functions and their Graphs |
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| f is a function from set A to a set B if each element x in A can be associated with a unique element in B. |
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| The unique element B which f associates with x in A denoted by f (x). |
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| In the above definition of the function, set A is called domain. |
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| In the above definition of the function, set B is called co-domain. |
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A real valued function f : A to B or simply a real function 'f ' is a rule which associates to each possible real number x A, a unique real number f(x) B, when A and B are subsets of R, the set of real numbers. |
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| In other words, functions whose domain and co-domain are subsets of R, the set of real numbers, are called real valued functions. |
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| If 'f ' is a function and x is an element in the domain of f, then image |
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| f(x) of x under f is called the value of 'f ' at x. |
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| Constant function |
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| A function f : A ®
B Such that A, B Ì R, is said to be a constant
function if there exist K Î B such that f(x) = k. |
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| Domain = A |
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| Range = {k} |
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| The graph of this function is a line or line segment parallel to x-axis. Note that, if k>0, the graph B is above X-axis. If k<0, the graph is below the x-axis. If k = 0, the graph is x-axis itself. |
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| Identity function |
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| A function f : R®
R is said to be an identity function if for all x Î
R, f(x) = x. |
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| Domain = R |
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| Range = R |
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| Polynomial function |
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| A function f : R®
R is said to be a polynomial function if for each x Î
R, f(x) is a polynomial in x. |
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| f(x) = x3 + x2 + x |
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| Modulus function |
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f : R ® R such
that f(x) = |x|,  is called the modulus function or absolute value function. |
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| Domain = R |
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| Square root function |
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| Since square root of a negative number is not real, we define a
function f : R+ ® R such that |
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| Greatest integer function or Step function (Floor function) |
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| f (x) = [x] = greatest integer less than or equal to x |
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[x] = n, where n is an integer such that  |
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| Smallest integer function (Ceiling function) |
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For a real number x, we denote by [x], the smallest integer greater than or equal to x. For example, [5 . 2] = 6, [-5 . 2] = -5, etc. The function f:R R defined by |
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f(x) = [x], x R |
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| is called the smallest integer function or the ceiling function. |
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| Domain: R |
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| Range : Z |
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| Exponential function |
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| The exponential function is defined as f(x) = ex. Its graph is |
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| Logarithmic function |
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| Logarithmic function is f (x) = log x. Its graph is |
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| Trigonometric functions |
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| Trigonometric functions are sinx, cosx, tanx, etc. The graph of these functions have been done in class XI. |
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| Inverse functions |
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| Inverse functions are sin-1x, cos-1x, tan-1x etc. The graph of these functions have been done in class XI. |
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| Signum functions |
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| Odd function |
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A function f : A B is said to be an odd function if |
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f(x) = - f(-x) for all x A |
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| The domain and range of f depends on the definition of the function. |
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| Examples of odd function are |
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| y = sinx, y = x3, y = tanx |
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| Even function |
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A function f : A B is said to be an even function if |
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f(x) = f(-x) for all x A. |
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| The domain and range of f depends on the definition of the function. |
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| Examples of even function are |
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| y = cosx, y = x2, y = secx |
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| A polynomial with only even powers of x is an even function. |
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| Reciprocal function |
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