Graph natural log of x^2+1

Problem

y=ln(x2+1)

Solution

1. Identify the domain, which consists of all real numbers because x2+1≥1 for all x meaning the argument of the natural logarithm is always positive.

2. Determine the symmetry by substituting −x for x which gives ln((−x)2+1)=ln(x2+1) indicating the function is even and symmetric about the yaxis.

3. Find the intercepts by setting x=0 to get y=ln(1)=0 so the graph passes through the origin (0,0)

4. Analyze the derivative to find extrema by calculating d(y)/d(x)=(2*x)/(x2+1) The derivative is zero at x=0 negative for x<0 (decreasing), and positive for x>0 (increasing), confirming a global minimum at (0,0)

5. Examine the end behavior as x→∞ or x→−∞ Since x2+1 grows without bound, ln(x2+1) also approaches ∞ though it grows very slowly.

6. Determine concavity by finding the second derivative d2(y)/(d(x)2)=(2*(1−x2))/((x2+1)2) The graph is concave up for −1<x<1 and concave down for |x|>1 with inflection points at x=1 and x=−1

Final Answer

y=ln(x2+1)

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