1 Given y = ln x. 1.1.1 Sketch the graph. 1.1.2 State the range of the graph mentioned. 1.1.3 Is the graph continuous or discontinuous? 1.1.4 Is the graph a function or relation? 1.1.5 Write down the equation of the inverse of the graph mentioned in question1.1

• The domain of $y = \ln x$ is $x > 0$.
• The graph passes through the point $(1, 0)$ because $\ln 1 = 0$.
• There is a vertical asymptote at $x = 0$ (the y-axis).
• As $x$ approaches 0 from the right, $y$ approaches $-\infty$.
• As $x$ approaches $\infty$, $y$ approaches $\infty$.

To sketch the graph, draw a coordinate plane. Draw a dashed vertical line at $x=0$ (the y-axis) to represent the asymptote. Plot the point $(1,0)$. Draw a smooth curve that starts from near the bottom of the y-axis (approaching $x=0$) and passes through $(1,0)$, then continues upwards and to the right, increasing slowly.

1.1.2 State the range of the graph mentioned.

As $x$ varies over its domain $(0, \infty)$, the value of $\ln x$ can take any real number.

The range of $y = \ln x$ is all real numbers.

1.1.3 Is the graph continuous or discontinuous?

A function is continuous if its graph can be drawn without lifting the pen. The natural logarithm function has no breaks, jumps, or holes within its domain $x > 0$.

The graph of $y = \ln x$ is continuous over its domain.

1.1.4 Is the graph a function or a relation?

For every valid input $x$ in the domain ($x > 0$), there is only one corresponding output $y$. A vertical line drawn anywhere in the domain $x > 0$ will intersect the graph at most once.

The graph of $y = \ln x$ is a function.

1.1.5 Write down the equation of the inverse of the graph mentioned in question 1.1

Given $y = \ln x$.
Swap variables: $x = \ln y$.

To isolate $y$, convert the logarithmic equation to an exponential equation. The natural logarithm $\ln y$ is $\log_e y$.
So, $x = \log_e y$ implies $y = e^x$.

The equation of the inverse is: