We start with the assumption f(x) = f(y). Then
\({x \over x^2-1}={y \over y^2-1}\)
\(\Rightarrow x(y^2-1)=(x^2-1)\)
\(\Rightarrow xy^2-x-yx^2+y-0\Rightarrow (y-x)(xy+1)=0\)
This implies that x =y or xy = - 1. So if we select two numbers x and y so that xy = -1, then f(x) = f(y).  \(f(x)=f(y) \cdot\left(2,-\frac{1}{2}\right),\left(7,-\frac{1}{7}\right),\left(-2, \frac{1}{2}\right)\)are some among the infinitely many possible pairs. Thus \(f(2)=f\left(\frac{-1}{2}\right)=\frac{2}{3}\). That is, f(x) = f(y) does not imply x = y. Hence it is not one-to-one.

f(x) = |x| − 1 causes the graph of the functionf(x) = |x| shifts to the downward for one unit. f(x) = |x| + 1 causes the graph of the function f(x) = |x| shifts to the upward for one unit.

The graphs figures (i) and (ii) look identical until we compare the scales on the y-axis. The scale in figure(ii) is four times as large, reflecting the multiplication of the original function by 4 (i).
The effect looks different when the functions are plotted on the same scale as in figure(iii).
The graph of y = (4x)² - 1 is shown in figure (iv). Can you spot the difference between figure (i) and figure (iv)? In this case, x-scale has now changed, by the same factor of 4 as in the function (figure (iv)).
To see this, note that substituting \(x={1\over 4}\) into (4x)² - 1 produces 12 - 1, 1, exactly the same as substituting x = 1 into the original function (figure (i)), When plotted on the same set of axes (as in figure (v» the parabola y = (4x)² - 1 looks thinner.
Here, the x-intercepts are different, but y-intercepts are the same.

f(x) = x² + 1causes the graph of the function f(x) = x² shifts to the upward for one unit.
f(x) = (x + 1)² causes the graph of the function f(x) = x² shifts to the left for one unit.

\(f(x)={1\over 2}x^2\) causes the graph of the function \(f(x)=x^2\) stretches towards x-axis since the multiplying factor is \({1\over 2}\) which is less than one. \(f(x)=2x^2\)causes the graph of the function \(f(x)=x^2\) compresses towards the y-axis that is, moves away from the x-axis since the multiplying factor is 2 which is greater than one.

For the curve \(f(x)=\sqrt{x},\) we have \(f(-x)=\sqrt{-x}\) and hence \(f(-x)=\sqrt{-x}\) where x < 0, is the reflection of \(f(x)=\sqrt{x}\) about y-axis.

(g o f) (x) = g(f(x)) = g(3x - 4) = (3x-4)²+3 = 9x²-24x+19
(f o g)(x) = f(g(x)) = f(x²+ 3) = 3(x²+ 3)-4 = 3x²+5.
Thus the operation "Composition of functions" is in general. (not commutative)