Questions: Graph the logarithmic function. g(x)=-3+log base 1/3 of x Plot two points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button.

Graph the logarithmic function. g(x)=-3+log base 1/3 of x Plot two points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button.
Transcript text: Graph the logarithmic function. \[ g(x)=-3+\log _{1 / 3} x \] Plot two points on the graph of the function, and also draw the asymptote. Then click on the graph-a-function button.
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Solution

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Solution Steps

Step 1: Find the vertical asymptote.

The function g(x) = -3 + log₁⸝₃(x) has a vertical asymptote where the argument of the logarithm is equal to zero. In this case, the argument is simply x. Thus the vertical asymptote is x = 0.

Step 2: Identify two points.

We can choose any two x-values greater than 0 since the domain of g(x) is x > 0. It will be helpful to choose values of x which simplify the log expression.

  • If x = 3, then g(3) = -3 + log₁⸝₃(3) = -3 + (-1) = -4. So the point (3, -4) is on the graph.
  • If x = 1, then g(1) = -3 + log₁⸝₃(1) = -3 + 0 = -3. So the point (1, -3) is on the graph.
Step 3: Plot the points and the asymptote.

Plot the two points (3, -4) and (1, -3) on the graph. Draw the vertical asymptote at x = 0. Sketch the curve of the logarithmic function, approaching the asymptote as x approaches 0 from the right, and passing through the two plotted points.

Final Answer:

The graph should have a vertical asymptote at x=0, and pass through the points (1,-3) and (3,-4). It should be a decreasing logarithmic graph since the base is between 0 and 1.

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