Question

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)?

vertical asymptote of (x = -4)

range of ((8, \infty))

domain of ((4, \infty))

y-intercept at ((0, 10))

x-intercept at ((1, 0))

Explanation:

Analyze the parent function and its features

The parent function is:

$$ f(x) = \log_2 x $$

Key features of \(f(x)\):

• Domain: \((0, \infty)\)
• Range: \((-\infty, \infty)\)
• Vertical asymptote: \(x = 0\)
• \(x\)-intercept: \((1, 0)\)
• \(y\)-intercept: None

Apply the transformations to find the features of g(x)

The transformed function is:

$$ g(x) = f(x + 4) + 8 = \log_2(x + 4) + 8 $$

This represents a horizontal shift left by 4 units and a vertical shift up by 8 units.

• New vertical asymptote: \(x = 0 - 4 \implies x = -4\)
• New domain: \((-4, \infty)\)
• New range: \((-\infty, \infty)\)
• New \(y\)-intercept:

$$ g(0) = \log_2(0 + 4) + 8 = \log_2(4) + 8 = 2 + 8 = 10 \implies (0, 10) $$

Evaluate the given options

• "vertical asymptote of \(x = -4\)": True.
• "range of \((8, \infty)\)": False (the range of any logarithmic function of this form remains \((-\infty, \infty)\)).
• "domain of \((4, \infty)\)": False (the domain is \((-4, \infty)\)).
• "y-intercept at \((0, 10)\)": True.
• "x-intercept at \((1, 0)\)": False (the new \(x\)-intercept is where \(\log_2(x+4) + 8 = 0 \implies x+4 = 2^{-8} \implies x = -4 + \frac{1}{256}\)).

Answer:

• vertical asymptote of \(x = -4\) (Correct answer)
• range of \((8, \infty)\)
• domain of \((4, \infty)\)
• y-intercept at \((0, 10)\) (Correct answer)
• x-intercept at \((1, 0)\)