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The logit function is

$ ln \frac{F(x)}{1-F(x)} $. According to wikipedia and the variety of materials that I have read, we can write this in linear form ($\beta_0 + \beta_1x$) for purposes of logistic regression. https://en.wikipedia.org/wiki/Logistic_regression

However, the graph of this function only looks linear between $.1

https://en.wikipedia.org/wiki/Logit#/media/File:Logit.svg

When the texts write this as a linear equation, are they simply approximating and assuming that p is between .1 and .9, where it is mostly linear?

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    In lr, $p$ is actually represented like $\frac{e^{(\beta_0+\beta_1 x)}}{e^{(\beta_0+\beta_1 x)} + 1}$. – 2017-02-23
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    But that depends on writing the logit function in linear form. The logit function only appears linear in the middle values, Why can we still write it in linear form? – 2017-02-23
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    In classification problem, we usually try to find the posterior probability $P(y|x)$, and assign x to the most probable class. So in LR, we want to find a model that have linear boundary, logistic function $f(x)=\frac{1}{1+e^{\beta^T x}}$ is choosed to model the posterior probability, since $f(x)\in (0, 1)$. And then we find the logit function of $f(x)$ can be written in linear form, not approximated. – 2017-02-24

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