How to graph \(\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}-{\cos{{\left({2}{x}-{2}\right)}}}}}{{{\left({x}-{1}\right)}^{{2}}}}}\) analytically? I only have

Mary Bates

Mary Bates

Answered question

2022-04-06

How to graph f(x)=1cos(2x2)(x1)2 analytically?
I only have an issue when it comes to determining the hole at x=1 analytically. Graphing the function on a graphing utility reveals a hole at x=1 rather than a vertical asymptote. I know that by simply plugging in the value 1 for x, one will see that both the numerator and denominator equate to zero. This shows that there is at least one factor of (x-1) that can be extracted from the numerator.
I would like to know how to simplify this function analytically in such a way that two factors of (x-1) can be obtained from the numerator. I believe one must make use of an inverse function down the line in order to extract the obvious (x-1) hiding in the cos(2x2)=cos(2(x1)) term. That is as far as I got in the problem.

Answer & Explanation

gigglesbuggk1co

gigglesbuggk1co

Beginner2022-04-07Added 13 answers

Hint: look at f(x+1)=1cos2xx2=2sin2xx2 then shift the origin back.

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