The functions defined have domain R Column 1 (a) 7 x+1 (b) cosx (c) sinx (d) 1+logx Column 2 (p)

step 1 Hi, in the question, we are given some functions in the column one that are defined the domain R

The functions defined have domain R Column 1 (a) 7 x+1 (b)...

The functions defined have domain $\mathrm{R}$
Column 1
(a) $7 x+1$
(b) $\cos x$
(c) $\sin x$
(d) $1+\log x$

Column 2
$(\mathrm{p})$ onto $[-1,1]$ but not one-one $[0, \pi]$
(q) onc-one on $[0, \pi]$ but not onto $\mathbf{R}$
(r) onc-one and onto $\mathbf{R}$
(s) one-one on $(0, \infty)$

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are defined on positive real numbers. They are strictly increasing and therefore one-to-one on their natural domains, but their domain must be carefully considered since logarithms are not defined for non-positive values.

Trigonometric Functions

Trigonometric functions such as sine and cosine are periodic and oscillatory in nature, meaning they repeat values over intervals. Although they have a fixed range, they are not one-to-one on their entire domains due to their periodic behavior, which often necessitates restricting the domain to restore injectivity.

Linear Functions

Linear functions, generally represented by expressions of the form ax + b, are characterized by constant rates of change and produce straight-line graphs. Their simplicity typically ensures that they are both injective and surjective when considered over the real numbers.

Domain Restriction for Invertibility

Sometimes functions that are not one-to-one over their entire natural domain can be made invertible by restricting the domain to an interval where they are monotonic. This technique is used to define inverse functions for functions like the sine and cosine, which are otherwise not injective when taken over all real numbers.

Injective (One?to?One) Functions

An injective function maps each element of its domain to a unique element in its range, meaning no two different inputs produce the same output. This property is important for ensuring that a function has an inverse, at least on the interval where it is one?to?one.

Function Domain and Range

Every function is defined by a set of inputs called its domain and a set of possible outputs called its range. Understanding these sets is essential in determining how the function behaves and what values it can take, particularly when comparing functions or altering their domains in order to achieve desired properties.

Surjective (Onto) Functions

A surjective function covers the entire target set, meaning every element in the codomain is an output of the function from some input in the domain. Surjectivity ensures that the function's mapping is comprehensive over its specified range.

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