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 function’s domain and range. h(x) = log x - 5 graph h(x) = log x - 5. 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 domain is \( (0, \infty) \), range is \( (-\infty, \infty) \), and vertical asymptote is \( x = 0 \).

Step2: Identify Transformation

For \( h(x)=\log x - 5 \), this is a vertical shift down by 5 units of \( f(x)=\log x \). Vertical shifts do not affect the vertical asymptote or the domain (since we're not shifting horizontally or reflecting/ stretching in a way that changes the domain of the log function).

Step3: Determine Asymptote

Since vertical shifts don't change the vertical asymptote of a logarithmic function, the vertical asymptote of \( h(x) \) is the same as \( f(x) \), which is \( x = 0 \).

Step4: Determine Domain and Range

• Domain: The argument of the log function \( x \) must be positive, so domain is \( (0, \infty) \).
• Range: A vertical shift down by 5 units doesn't change the range of a logarithmic function (it still spans all real numbers), so range is \( (-\infty, \infty) \).

Answer:

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

(For graphing: Start with the graph of \( y = \log x \), then shift every point down 5 units. The vertical asymptote \( x = 0 \) (dashed line) remains the same.)