If \(f:R-\{ -1,1\}\rightarrow R\) is defined by \(f(x)={x \over x^2-1},\) verify whether f is one-to-one or not.

Let f and g be the two functions from R to R defined by f(x) = 3x - 4 and g(x) = x²+ 3. Find g o f and f o g.

Consider the positive branches y² = x and y² = -x.

Consider the functions: 
i) \(f(x)=x^2,\)
ii) \(f(x)={1\over 2}x^2,\)
iii) \(f(x)=2x^2\)

Consider the functions:
i) \(f(x)=x^2,\)
ii) \(f(x)=x^2+1,\)
iii) \(f(x)={(x+1)}^{2}\)

Compare and contrast the graph y = x² - 1, y = 4(x² - 1) and y = (4x)² = 1.

By using the same concept applied in previous example, graphs of y = sin x and y = sin 2x, and also their combined graphs are given figures (a), (b) and (c). The minimum and maximum values of sin x and sin 2x are the same. But they have different x-intercepts. The x-intercepts for y = sin x are \(\pm n\pi\) and for y = sin 2x are \(\pm{1\over 2}n\pi,\ n\in Z.\) ​​​

Consider the functions:
(i) f(x) = |x|
(ii) f(x) = |x| − 1
(iii) f(x) = |x| + 1