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Even and odd functions are related to symmetry of functions. The symmetry of a function is visualized by the planar plot of a function, which may show symmetry with respect to either an axis (y-axis) or origin.
Since functions need not always be symmetric, they may neither be even nor be odd. The parity of a function i.e. whether it is even or odd is determined with certain algebraic algorithm. Further, symmetry of functions may change subsequent to mathematical operations.
The values of even function at x=x and x=-x are same.
$$f\left(-x\right)=f\left(x\right)$$
An even function is symmetric about y-axis. If we consider the axis as a mirror, then the plot in first quadrant has its mirror image (bilaterally inverted) in second quadrant. Similarly, the plot in fourth quadrant has its mirror image (bilaterally inverted) in third quadrant.
Some examples of even functions are ${x}^{2},\left|x\right|\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\mathrm{cos}x$ . In each case, we see that :
$$\Rightarrow f\left(-x\right)={\left(-x\right)}^{2}={x}^{2}=f\left(x\right)$$
$$\Rightarrow f\left(-x\right)=|-x|=\left|x\right|=f\left(x\right)$$
$$\Rightarrow f\left(-x\right)=\mathrm{cos}\left(-x\right)=\mathrm{cos}x=f\left(x\right)$$
The right side is mirror image of left hand side and the left side is mirror image of right hand side of the curve.
It is important to see that if we rotate the curve by 180° about y-axis, then the appearance of the rotated curve is same as the original curve. We can state this alternatively as : if we rotate left hand side of the curve by 180° about y-axis, then we get the right hand curve and vice-versa.
Problem 1: Prove that the function f(x) is “even”, if
$$f\left(x\right)=x\frac{{a}^{x}-1}{{a}^{x}+1}$$
Solution : For function being “even”, we need to prove that :
$$f\left(-x\right)=f\left(x\right)$$
Here,
$$\Rightarrow f\left(-x\right)=-x\frac{{a}^{-x}-1}{{a}^{-x}+1}=-x\frac{\frac{1}{{a}^{x}}-1}{\frac{1}{{a}^{x}}+1}$$
$$\Rightarrow f\left(-x\right)=-x\frac{\frac{1-{a}^{x}}{{a}^{x}}}{\frac{1+{a}^{x}}{{a}^{x}}}=-x\frac{1-{a}^{x}}{1+{a}^{x}}$$
$$\Rightarrow f\left(-x\right)=x\frac{{a}^{x}-1}{{a}^{x}+1}=f\left(x\right)$$
Problem 2: If an even function “f” is defined on the interval (-5,5), then find the real values for which
$$f\left(x\right)=f\left(\frac{x+1}{x+2}\right)$$
Solution : It is given that function “f” is even. Hence, arguments of the functions on two sides are related either as
$$\Rightarrow x=\frac{x+1}{x+2}$$
or as :
$$\Rightarrow x=-\frac{x+1}{x+2}$$
From the first relation,
$$\Rightarrow {x}^{2}+x-1=0$$
$$\Rightarrow x=\frac{-1\pm \sqrt{5}}{2}$$
From the second relation,
$$\Rightarrow {x}^{2}+3x+1=0$$
$$\Rightarrow x=\frac{-3\pm \sqrt{5}}{2}$$
We see that values are within the specified domain. Hence, all the four solutions satisfy the given equation.
The values of odd function at x=x and x=-x are equal in magnitude but opposite in sign.
$$f\left(-x\right)=-f\left(x\right)$$
An odd function is symmetric about origin of the coordinate system. The plot in first quadrant has its mirror image (bilaterally inverted) in third quadrant. Similarly, the plot in second quadrant has its mirror image (bilaterally inverted) in fourth quadrant.
Some examples of odd functions are : $x,{x}^{3}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\mathrm{sin}x$ . In each case, we see that :
$$\Rightarrow f\left(-x\right)=-x=-f\left(x\right)$$
$$\Rightarrow f\left(-x\right)={\left(-x\right)}^{3}=-{x}^{3}=-f\left(x\right)$$
$$\Rightarrow f\left(-x\right)=\mathrm{sin}\left(-x\right)=-\mathrm{sin}x=-f\left(x\right)$$
The upper curve of these functions is exactly same as the lower curve across x-axis.
It is important to see that if we rotate the curve by 180° about origin, then the appearance of the rotated curve is same as the original curve. In other words, if we rotate right hand side of curve by 180° about origin, then we get left side of the curve. Further, it is interesting to note that we obtain left hand part of the plot of odd function in two steps : (i) drawing reflection (mirror image) of right hand plot about y-axis and (ii) drawing reflection (mirror image) of “reflection drawn in step 1” about x-axis.
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