Question

begin by graphing f(x) = log x. use transformations of this graph to graph the given function. graph and give the equation of the asymptote. use the graphs to determine the functions domain and range. h(x) = log x - 3 graph h(x) = log x - 3. graph the asymptote of h(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 \( f(x)=\log x \). Its vertical asymptote is \( x = 0 \) (y - axis), domain is \( (0,\infty) \), range is \( (-\infty,\infty) \).

Step2: Identify Transformation

For \( h(x)=\log x - 3 \), this is a vertical shift down by 3 units of \( f(x)=\log x \). Vertical shifts do not affect the vertical asymptote, domain (since we are not shifting horizontally or reflecting over y - axis or changing the argument of the log in a way that affects the domain).

Step3: Find Asymptote, Domain, Range

• Asymptote: Since vertical shift doesn't change the vertical asymptote, the equation of the vertical asymptote for \( h(x) \) is \( x = 0 \).
• Domain: The domain of a logarithmic function \( y=\log(b(x - h))+k \) (here \( b = 1 \), \( h = 0 \), \( k=-3 \)) is determined by the argument \( x>0 \), so domain is \( (0,\infty) \).
• Range: Vertical shifts do not change the range of a logarithmic function. The range of \( \log x \) is \( (-\infty,\infty) \), so the range of \( h(x)=\log x-3 \) is also \( (-\infty,\infty) \).

Answer:

• Asymptote Equation: \( x = 0 \)
• Domain: \( (0,\infty) \)
• Range: \( (-\infty,\infty) \)