- #1

RidiculousName

- 28

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I recently had to find what \(\displaystyle f(7)\) equals if \(\displaystyle f(x) = \frac{x^2-11x+28}{x-7}\). I first tried \(\displaystyle \frac{x^2-11x+28}{x-7} \cdot \frac{x+7}{x+7}\), and it seemed like a perfect fit since I eventually got to \(\displaystyle \frac{x^2(x-4)-49(x+4)}{x^2-49}=(x-4)(x+4)\), but that gave me \(\displaystyle f(7)=33\), instead of the right answer which was \(\displaystyle f(7)=3\) since \(\displaystyle \frac{x^2-11x+28}{x-7}=\frac{(x+7)(x-4)}{x-7}=x-4\).

I don't have much experience with finding discontinuities, and I am confused because I had thought multiplying by by the conjugate was always the right way to go with these problems. Since I was wrong about that, I want to know how to tell what the wrong processes are with these types of problems so that I can do them correctly in the future. Is there a way I could've known that trying to multiply by \(\displaystyle \frac{x+7}{x+7}\) was the wrong approach here?

I don't have much experience with finding discontinuities, and I am confused because I had thought multiplying by by the conjugate was always the right way to go with these problems. Since I was wrong about that, I want to know how to tell what the wrong processes are with these types of problems so that I can do them correctly in the future. Is there a way I could've known that trying to multiply by \(\displaystyle \frac{x+7}{x+7}\) was the wrong approach here?

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