Question

begin by graphing f(x) = \log_3 x. then use transformations of this graph to graph the given function. what is the vertical asymptote? use the graphs to determine the given functions domain and range.

g(x) = \log_3 (x + 5)

graph the function g(x) = \log_3 (x + 5). 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 parent function

The parent function is \( f(x)=\log_3 x \). Its vertical asymptote is \( x = 0 \), domain is \( (0,+\infty) \), range is \( (-\infty,+\infty) \).

Step2: Determine transformation

The function \( g(x)=\log_3(x + 5) \) is a horizontal shift of \( f(x) \). For \( y=\log_b(x + h) \), it's a shift left by \( h \) units (when \( h>0 \)). Here \( h = 5 \), so shift \( f(x) \) left 5 units.

Step3: Find vertical asymptote

For \( f(x)=\log_3 x \), asymptote is \( x = 0 \). After shifting left 5 units, the vertical asymptote of \( g(x) \) is \( x=-5 \).

Step4: Find domain

The argument of log must be positive: \( x + 5>0\implies x>-5 \), so domain is \( (-5,+\infty) \).

Step5: Find range

Logarithmic functions have range \( (-\infty,+\infty) \), and horizontal shifts don't change the range. So range of \( g(x) \) is \( (-\infty,+\infty) \).

Answer:

Vertical Asymptote: \( x = -5 \)
Domain: \( (-5, \infty) \)
Range: \( (-\infty, \infty) \)