Question

begin by graphing f(x) = log₂x. then use transformations of this graph to graph the given function. what is the vertical asymptote? use the graphs to determine the given function’s domain and range. g(x) = log₂(x + 8) graph the function g(x) = log₂(x + 8). graph the asymptote of g(x) as a dashed line. use the graphing tool to graph the function. click to enlarge graph

Explanation:

Step1: Analyze the parent function

The parent function is \( f(x)=\log_2 x \). The vertical asymptote of \( f(x) \) is \( x = 0 \), and its domain is \( (0,+\infty) \), range is \( (-\infty,+\infty) \).

Step2: Determine the transformation

The function \( g(x)=\log_2(x + 8) \) is a horizontal shift of \( f(x) \). For a function \( y=\log_b(x + h) \), it is a shift of \( y = \log_b x \) left by \( h \) units (when \( h>0 \)). Here, \( h = 8 \), so we shift \( f(x)=\log_2 x \) left by 8 units.

Step3: Find the vertical asymptote

For the parent function \( f(x)=\log_2 x \), the vertical asymptote is \( x = 0 \). After shifting left by 8 units, the vertical asymptote of \( g(x) \) becomes \( x=-8 \).

Step4: Determine the domain

The argument of the logarithm must be positive. So for \( g(x)=\log_2(x + 8) \), we need \( x + 8>0 \), which gives \( x>-8 \). So the domain is \( (-8,+\infty) \).

Step5: Determine the range

Since horizontal shifts do not affect the range of logarithmic functions, the range of \( g(x) \) is the same as the range of \( f(x) \), which is \( (-\infty,+\infty) \).

Answer:

• Vertical Asymptote: \( x=-8 \)
• Domain: \( (-8,+\infty) \)
• Range: \( (-\infty,+\infty) \)