A salesperson whose annual earnings can be represented by the function A(x) = 30,000 + 0.04x, where x is the rupee value of the merchandise he sells. His son also in sales and his earnings are represented by the function S(x) = 25,000 + 0.05x. Find (A+S) (x) and determine the total family income if they each sell Rs. 1,50,00,000 worth of merchandise.

A simple cipher takes a number and codes it, using the function f(x) = 3x - 4. Find the inverse of this function, determine whether the inverse is also a function and verify the symmetrical property about the line y = x(by drawing the lines)

From the curve y = sin x, graph the functions.
(i) y = sin(-x)
(ii) y = -sin(-x)
(iii) ​\(y=sin\left( {\pi\over 2}+x\right)\) which is cos x​
(iv) ​\(y=sin\left({\pi\over 2}-x \right)\)​ which is also cos x (refer trigonometry)

From the curve y = x, draw
(i) y = - x
(ii) y = 2x
(iii) y = x + 1
(iv) ​\(y={1\over 2}x+1\)​
(v) 2x + y + 3 = 0

Write the values of f at -3, 5, 2, -1, 0 if
\(f(x)=\begin{cases} x^2+x-5\quad if\ x \in(-\infty, 0) \\x^2+3x-2\quad if\ x\in(3,\infty) \\x^2\quad \quad \quad \quad \quad if\ x\ \in(0,2) \\x^2-3 \quad \quad \quad otherwise \end{cases}\)

Find the range of the function \(\frac { 1 }{ 2cosx-1 } \)

If f:R \(\rightarrow\) R is defined by f(x) = 3x - 5, prove that f is a bijection and find its inverse.

In the set Z of integers, define mRn if m - n is a multiple of 12. Prove that R is an equivalence relation.

Check the following functions for one-to-oneness and ontoness.
(i) \(f:N\rightarrow N\) defined by f(n) = n².
(ii) \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by f(n) = n².

Find the range of the function \(f(x)={1 \over 1-3\cos\ x}.\)