Question

begin by graphing ( f(x)=log _{4}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.
( h(x)=4+log _{4}x )
graph the function ( h(x)=4+log _{4}x ). graph the asymptote of ( h(x) ) as a dashed line. use the graphing tool to graph the function.

Explanation:

Step1: Analyze the transformation

The function \(h(x)=4 + \log_{4}x\) is a vertical shift of \(f(x)=\log_{4}x\) up by 4 units.

Step2: Find the vertical asymptote

For \(y = \log_{a}x\), the vertical asymptote is \(x = 0\). Since \(h(x)=4+\log_{4}x\) is a vertical shift of \(y=\log_{4}x\), the vertical asymptote remains \(x = 0\).

Step3: Determine the domain

The domain of \(y=\log_{a}x\) is \(x>0\). A vertical shift does not affect the domain. So the domain of \(h(x)\) is \((0,\infty)\).

Step4: Determine the range

The range of \(y=\log_{a}x\) is \((-\infty,\infty)\). A vertical shift of 4 units up changes the range to \((-\infty,\infty)\) (because we are just adding 4 to each output value of \(\log_{4}x\), and \(\log_{4}x\) can take any real - valued output).

Answer:

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