How to plot function of the form $f(x)= \dfrac{x+1}{2x^2+1}$ manually
What are the functions of this type called? Is it a quadratic function?
Edit ..Thanks for letting me know that functions of this form are called rational functions
Please help me. I am new to this.
Hint for the plot: Here are some things one typically looks for:
Intersections with axes. To obtain these, we solve:
$$f(0)=y, \text{ and } f(x)=0,$$ for $x$ and $y$ respectively.
2.1 Local monotonicity. For this one simply takes the derivative of $f$ with respect to $x$ and determines the sub-domains in which the function increases/decreases.
2.2. Turning points . Points in which $f'(x)=0$ and such that $f'(x)$ "changes sign" (thanks @Henry).
Local convexity/concavity. For that, we study the second derivative of $f$ with respect to $x$ and solves the inequalities $f''(x)<0$ and $f''(x)>0$
Limits. Compute $\lim_{x\to \infty}f(x)$ and $\lim_{x\to -\infty}f(x)$.
Discontinuities. Look for possible discontinuities in the domain of $f$.
Can you determine these for your function of interest?