Graph f(x)=- natural log of x-1+3

Problem

ƒ(x)=−ln(x−1)+3

Solution

1. Identify the parent function and its basic properties. The parent function is y=ln(x) which has a vertical asymptote at x=0 and passes through the point (1,0)

2. Determine the horizontal shift by looking at the argument of the logarithm. The term (x−1) indicates a shift to the right by 1 unit. The vertical asymptote moves from x=0 to x=1

3. Apply the reflection across the x-axis. The negative sign in front of the natural log, −ln(x−1) reflects the graph vertically, changing the increasing curve into a decreasing curve.

4. Determine the vertical shift by looking at the constant term. The +3 indicates a shift upward by 3 units.

5. Find key points to plot the graph. When x=2 ƒ(2)=−ln(2−1)+3=−ln(1)+3=0+3=3 When x=e+1 ƒ*(e+1)=−ln(e)+3=−1+3=2

6. Sketch the curve starting from the vertical asymptote x=1 passing through (2,3) and (e+1,2) and approaching negative infinity as x increases.

Final Answer

ƒ(x)=−ln(x−1)+3

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