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 - 6

graph h(x) = log x - 6. graph the asymptote of h(x) as a dashed line. use the graphing tool to graph the function.
click to enlarge graph
what is the vertical asymptote of h(x)?
(type an equation.)
what is the domain of h(x) = log x - 6?
(simplify your answer. type your answer in interval notation.)
what is the range of h(x) = log x - 6?
(simplify your answer. type your answer in interval notation.)

Explanation:

Step1: Analyze Vertical Asymptote

For the parent function \( f(x)=\log x \), the vertical asymptote is \( x = 0 \) (since the logarithm function \( \log x \) is undefined at \( x = 0 \) and approaches \( -\infty \) as \( x \to 0^+ \)). The transformation \( h(x)=\log x - 6 \) is a vertical shift down by 6 units. Vertical shifts do not affect the vertical asymptote. So the vertical asymptote of \( h(x) \) is still \( x = 0 \).

Step2: Determine Domain

The domain of a logarithmic function \( \log x \) is the set of all positive real numbers, i.e., \( x>0 \). A vertical shift does not change the domain because it only affects the \( y \)-values. So the domain of \( h(x)=\log x - 6 \) is \( (0, \infty) \) in interval notation.

Step3: Determine Range

The range of the parent function \( f(x)=\log x \) is all real numbers, \( (-\infty, \infty) \). A vertical shift down by 6 units shifts the graph vertically but does not restrict the \( y \)-values. So the range of \( h(x)=\log x - 6 \) is still \( (-\infty, \infty) \).

Answer:

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