Question

which equation is represented by the graph below?
options:
y = e^x
y = e^x - 1
y = ln x
y = ln x - 1

Explanation:

Step1: Analyze the domain and shape

The graph is defined for \( x > 0 \), which is the domain of the natural logarithm function \( y = \ln x \) (since exponential functions \( y = e^x \) are defined for all real \( x \)). So we can eliminate options with \( e^x \) (first two options).

Step2: Check the y-intercept or key points

For \( y=\ln x \), when \( x = 1 \), \( y=\ln(1)=0 \), which matches the graph passing through \( (1,0) \). For \( y=\ln x - 1 \), when \( x = 1 \), \( y=\ln(1)-1=-1 \), which does not match the graph (the graph passes through \( (1,0) \)). Also, the graph's vertical intercept (near \( x = 0^+ \)) and shape align with \( y=\ln x \) (the natural log function has a vertical asymptote at \( x = 0 \) and passes through \( (1,0) \), increasing slowly for \( x>1 \)).

Answer:

\( y = \ln x \) (the option corresponding to \( y=\ln x \))