Sketch a graph. Be sure to include asymptotes and significant points as needed. List the domain and range for each.  Equation: f(x) = ln(|1/2x|)+3  please provide detail explanation.

Expert Answers

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1.Determine for what x values the function is undefined.

Since 1/2x is undefined when x=0, there is an asymptote at x=0

2.Determine the x intercepts by equating to 0

`ln(|1/(2x)|)+3=0`

`ln(|1/(2x)|)=-3`

`|1/(2x)|=e^-3`

The first solution is:

`1/(2x)=e^-3`

`x=1/(2e^-3)=e^3/2=10.04`

The second solution is:

`-1/2x=e^-3`

`x=-e^3/2=-10.04`

The function is symmetric with respect to the x=0 axis.

Evaluate f(x) for a value of x between 0 and 10, say x=5.

f(x)=`ln(|1/(2*5)|)+3=-2.3+3=0.7`

Evaluate f(x) for a very high value, say x=`10^10`

f(x)=`ln(|1/(2x10^10)|)+3=5e-9+3=-19.11+3=-16.11`

The domain is x<0 and x>0

The range is (`-oo,oo)`

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