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)

Graph the function f(x) = x³ and \(g(x)=\sqrt[3]x\) on the same co-ordinate plane. Find f o g and graph it on the plane as well. Explain your results.

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 = sin x, draw y = sin |x|. (Hint: sin (-x) = -sin x)

Write the values of f at -4, 1, -2, 7, 0 if
\(f(x)=\left\{ \begin{matrix} -x+4& if -\infty <x\leq -3\\ x+4& if -3<x<-2\\ x^{2}-x& if -2\leq x < 1 \\ x-x^{2}& if 1\leq x<7\\ 0& otherwise\\ \end{matrix}\right.\)

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

Find all values of x that satisfies the inequality \({{2x-3}\over{(x-2)(x-4)}}<0.\)

Determine the region in the plane determined by the inequalities.
\(x-2y\ge 0,\ 2x-y\le -2,\ x\ge 0,\ y\ge 0.\)

If cosec \(\theta\) - sin \(\theta\) = a³ and sec \(\theta\) - cos \(\theta\) = b³, then prove that a²b² (a²+ b²) = 1

Prove that cos 5\(\theta\) = 16 cos² \(\theta\) - 20 cos³ \(\theta\) + 5 cos \(\theta\).

Solve the following equations cos \(\theta\) + cos 3\(\theta\) = 2 cos2\(\theta\)

Solve the following equations sin \(\theta\) + cos \(\theta\) = \(\sqrt { 2 } \)

If \(y=\frac{2\ sin\alpha}{1+cos\alpha+sin\alpha}\) then prove that \(\frac{1-cos\alpha+sin\alpha}{1+sin\alpha}=y\).

Using Heron's formula, show that the equilateral triangle has the maximum area for any fixed perimeter. [Hint: In xyz \(\le\)  k, maximum occurs when x = y = z]

By the principle of mathematical induction, prove that for n > 1
\(1^3 +2^3 +3^3 + .. +n^3=\left[n(n+1)\over 2\right]^2\)

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box, if atleast one black ball is to be included in the draw?

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2^nd hour, 4^th hour and n^th hour?

The normal boiling point of water is 100°C or 212°F· and the freezing point of water is 0 °C or 32°F.
(i) Find the linear relationship between C and F.
(ii) Find the value of C for 98.6°F and 
(iii) Find the value of F for 38°C.

Find the separate equation of the following pair of straight lines.
3x² + 2xy - y² = 0

If the pair of lines represented by x² - 2cxy -y² = 0 and x² - 2dxy -y² = 0 be such that each pair bisects the angle between the other pair, prove that cd = -1.

Consider a hollow cylindrical vessel, with circumference 24 cm and height 10 cm. An ant is located on the outside of vessel 4 cm from the bottom. There is a drop of honey at the diagrammatically opposite inside of the vessel, 3 cm from the top. 
(i) What is the shortest distance the ant would need to crawl to get the honey drop?
(ii) Equation of the path traced out by the ant.
(iii) Where the ant enter in to the cylinder? Here is a picture that illustrates the position of the ant and the honey.

Prove that\(\begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b &1 \\1 &1 &1+c \end{vmatrix}=abc\left( 1+{1\over a}+{1\over b}+{1\over c} \right) .\)

Solve the following problems by using Factor Theorem :
Solve \(\begin{vmatrix} 4-x & 4+x & 4+x \\ 4+x & 4-x & 4+x \\ 4+x & 4+x & 4-x \end{vmatrix}=0\) .

Sketch the graph of a function f that satisfies the given values :
f(0) is undefined
\(lim_{x\rightarrow0}f(x)=4\)
f(2) = 6
\(lim_{x\rightarrow2}f(x)=3\)

At the given point x_o discover whether the given function is continuous or discontinuous citing the reasons for your answer :\(x_{0}=3, f(x)= \begin{cases}\frac{x^{2}-9}{x-3}, & \text { if } x \neq 3 \\ 5, & \text { if } x=3\end{cases}\)