Questions: Participation Activity #19 This is similar to Try It #5 in the OpenStax text. Sketch a graph of f(x)=log3(x)+2 alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Participation Activity #19
This is similar to Try It #5 in the OpenStax text.
Sketch a graph of f(x)=log3(x)+2 alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Transcript text: Participation Activity \#19 This is similar to Try It \#5 in the OpenStax text. Sketch a graph of $f(x)=\log _{3}(x)+2$ alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Solution

Solution Steps

Step 1: Identify the Parent Function

The parent function of $ f(x) = \log_{3}(x) + 2 $ is $ g(x) = \log_{3}(x) $.

Step 2: Determine the Domain and Range

The domain of $ g(x) = \log_{3}(x) $ is $ (0, \infty) $.
The range of $ g(x) = \log_{3}(x) $ is $ (-\infty, \infty) $.

Step 3: Determine the Asymptote

The vertical asymptote of $ g(x) = \log_{3}(x) $ is $ x = 0 $.

Step 4: Transform the Parent Function

The function $ f(x) = \log_{3}(x) + 2 $ is a vertical shift of the parent function $ g(x) = \log_{3}(x) $ by 2 units upwards.

Final Answer

Domain of $ f(x) = \log_{3}(x) + 2 $: $ (0, \infty) $
Range of $ f(x) = \log_{3}(x) + 2 $: $ (-\infty, \infty) $
Vertical asymptote of $ f(x) = \log_{3}(x) + 2 $: $ x = 0 $

{"axisType": 3, "coordSystem": {"xmin": -1, "xmax": 5, "ymin": -2, "ymax": 5}, "commands": ["y = log(3, x)", "y = log(3, x) + 2"], "latex_expressions": ["$y = \\log_{3}(x)$", "$y = \\log_{3}(x) + 2$"]}

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