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Consider the graph of the function \( f(x) = \log_2 x \). What are the features of function \( g \) if \(...

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Consider the graph of the function \( f(x) = \log_2 x \).
What are the features of function \( g \) if \( g(x) = f(x + 4) + 8 \) ?
☐ domain of \((4, \infty)\)
☐ vertical asymptote of \( x = -4 \)
☐ range of \((8, \infty)\)
☐ \( x \)-intercept at \((1, 0)\)
☐ \( y \)-intercept at \((0, 10)\)

Answer

Correct Answers:

  1. Domain of \((4, \infty)\): True. The domain of \( g(x) = f(x + 4) + 8 \) is shifted 4 units to the left, so the domain starts at \( x = -4 \), which means \( x + 4 > 0 \Rightarrow x > -4 \).

  2. Vertical asymptote of \( x = -4 \): False. There is no vertical asymptote for logarithmic functions.

  3. Range of \((8, \infty)\): True. The range of \( f(x) \) is \((-\infty, \infty)\), and adding 8 shifts it to \((8, \infty)\).

  4. \( x \)-intercept at \((1, 0)\): False. The \( x \)-intercept is not at \((1, 0)\) for \( g(x) \).

  5. \( y \)-intercept at \((0, 10)\): False. The function \( g(x) \) does not have a \( y \)-intercept at \((0, 10)\).

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