Question

begin by graphing f(x)= ln 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) = ln (4x) graph h(x) = ln (4x). 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 the parent function

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

Step2: Identify the transformation

For \( h(x) = \ln(4x) \), we can use the property of logarithms: \( \ln(ab)=\ln a+\ln b \), so \( h(x)=\ln 4+\ln x \). This is a horizontal compression of the parent function \( f(x)=\ln x \) by a factor of \( \frac{1}{4} \) (since \( y = \ln(kx) \) for \( k>0 \) is a horizontal compression of \( y = \ln x \) by \( \frac{1}{k} \)).

Step3: Determine the asymptote

For logarithmic functions of the form \( \ln(kx) \) where \( k>0 \), the vertical asymptote occurs where the argument is zero. So, set \( 4x = 0 \), which gives \( x = 0 \). So the vertical asymptote is \( x = 0 \).

Step4: Determine domain and range

• Domain: The argument of the logarithm \( 4x>0 \), so \( x > 0 \), domain is \( (0, \infty) \).
• Range: The range of a logarithmic function (regardless of horizontal compressions/stretches or vertical shifts) is still \( (-\infty, \infty) \) because as \( x \) approaches \( 0^+ \), \( \ln(4x) \) approaches \( -\infty \), and as \( x \) approaches \( \infty \), \( \ln(4x) \) approaches \( \infty \).

Step5: Graphing (conceptual)

To graph \( h(x)=\ln(4x) \), start with the graph of \( f(x)=\ln x \). Then, horizontally compress it by a factor of \( \frac{1}{4} \). The vertical asymptote \( x = 0 \) remains (dashed line), and the graph passes through points like when \( x=\frac{1}{4} \), \( h(\frac{1}{4})=\ln(4\times\frac{1}{4})=\ln(1) = 0 \), so the point \( (\frac{1}{4}, 0) \) is on the graph, similar to how \( (1, 0) \) is on \( f(x)=\ln x \) but compressed horizontally.

Answer:

• Vertical Asymptote: \( x = 0 \)
• Domain: \( (0, \infty) \)
• Range: \( (-\infty, \infty) \)
• Graph: A horizontally compressed (by factor \( \frac{1}{4} \)) version of \( y = \ln x \) with vertical asymptote \( x = 0 \) (dashed line) and passing through \( (\frac{1}{4}, 0) \) etc. (When using the graphing tool, plot the asymptote \( x = 0 \) as a dashed line and plot points for \( h(x) \) like \( x=\frac{1}{4} \) ( \( y = 0 \) ), \( x = 1 \) ( \( y=\ln 4\approx1.386 \) ), \( x = \frac{1}{2} \) ( \( y=\ln(2)\approx0.693 \) ) and draw the curve approaching the asymptote \( x = 0 \) from the right and extending to \( \infty \) as \( x \) increases.)