Can you propose two functions of totally different expressions in x, but exactly the same numerical value when plotted?
I mean two functions; y1=f1(x), and y2=f2(x), which are totally of different expressions in x, but numerically of the same exact value, when plotted by a computer software, or when calculated at any point x=a, by hand. The functions could be assumed positive on the positive real-line, but this is not a binding restriction. You can also propose the functions only for x>0. None of the functions should be an infinite series, but they should be of finite mathematical terms.
For example f1(x) could contain trigonometric terms and f2(x) could contain exponential or polynomials. There should be some difference in the appearance of the functions you propose, but even some similar-looking but a bit different functions are also welcomed.
warning: I am not looking for trivial identities as:
sinh(x)=[exp(x)-exp(-x)] /2
Dear Saeb AmirAhmadi Chomachar
y1 = sinx and y2 = sin(x+a)
they have exactly the same graph but with different expressions.
But if y1 = f(x) = y2 = g(x) at any x, we should obtain an identity.
For example,
y1 = cos2 (x) , and y2 =(1+cos(2x))/2.
which is trivially true by our assumptions!
One may show strange relations, but in the end, it should be the same.
Best regards
Dear Issam Kaddoura
Thank you for the answer. But I had asked in my question, some discernibly different functions, at least in appearance, for example one should be trigonometric and the other exponential or polynomial, or both of the functions have these terms in common but with a somewhat different appearance. Your example is good, and I thank you again, but the difference in the expressions should be more conspicuous. y1 = sinx, and, y2 = sin(x+a) are both trigonometric with a phase shift.
Best regards.
Dear Saeb AmirAhmadi Chomachar
I think the attached pdf file includes an exciting example over
a discrete domain N.
Best wishes
Respected Michael Short and Issam Kaddoura
Thank you for the comments. I had already warned that I am not looking for trivial identities as that presented by Mr. Short.
And the recent example by Mr. Kaddoura is for discrete n or alternatively x, but I was looking for continuous functions in x.
I've attached a PDF to my comment here, as an example for the answer. It would be difficult to find such functions with such desired analogy. But I am still looking for a couple of functions with a bit more different shape, but neither very complicated nor with large expressions.
Update: in the PDF, the two expressions for f"(x) in terms of x, seem not exactly equal in some intervals (probably due to truncation error), but for a large interval they are exactly the same when I plotted them by Wolfram Mathematica.
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