Ch 15Functions — Class 11 Mathematics, concept-first.

A function (or mapping) from a set to a set , written , is a relation which associates to every element in a unique (exactly one) element in . We write , and say is the image of under .

Two independent properties refine what it means for a function to relate its domain and co-domain 'nicely'.

The very same function can be displayed in six different, completely equivalent ways. Using the running example 'the output exceeds twice the input by 1' as one fixed function throughout:

Graph of a function. When the domain of a function lies in , the function can be drawn as a curve in the -plane, consisting of every point with .

Value of a function. is called the value of the function at — simply the result of substituting into the formula for .

Some basic functions (all with unless stated otherwise).

3. Power functions. Form: , (a multiple of the -th power of ).

4. Polynomial function. is a polynomial function of degree , provided and every is real.

5. Radical function. Example: , .

6. Rational function. Definition: given polynomials , is defined for whenever . Example: , (Fig. 6.27). Domain: ; Range: .

7. Exponential function. Form: is an exponential function with base and exponent (or index) , where , , and . Example: and (Fig. 6.28). Domain: ; Range: .

8. Logarithmic function. Let ; we define if , for . This gives two equivalent forms: the logarithmic form and the exponential form — each converts directly to the other.

9. Change of base formula. For and : — a logarithm in one base can always be rewritten as a ratio of logarithms in any other convenient base .

9. Trigonometric function. The graphs of the trigonometric functions were already studied in Chapter 2 of Mathematics Book I; this section simply records their domain and range for reference: has doma…

Algebra of functions. Let and be functions with domains and respectively. Then the four combined functions , , , are all defined on the intersection (the set of inputs valid for both and at once), as…

Composition of functions. A method of combining a function with a function is composition, defined as , read ' composed with ' (Fig. 6.35).

Inverse functions. Let be a one-one and onto function, with for . The inverse function is defined by if (Fig. 6.36).

Piecewise defined functions. A function defined by two or more separate equations, each applying only on its own specified part of the domain, is called a piecewise defined function.

1. Signum function. Definition: is the piecewise function (Fig. 6.38). Domain: ; Range: .

Absolute value function (modulus function). Definition: is the piecewise function (Fig. 6.39). Domain: (or ); Range: .

3. Greatest integer function (step function). Definition: for every real , the greatest integer less than or equal to . is also called the floor function, sometimes written .

4. Fractional part function. Definition: for every real , , defined as — whatever is left over after removing the integer (floor) part.

Characteristic and mantissa of common logarithm . Since every real splits as (integer part plus fractional part), the same split applies to its logarithm: , where the integral part is called the chara…