begin by graphing f(x) = log x. use transformations of this graph to graph the given function. graph and…

Explanation:

Step1: Recall 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: Analyze Transformation for ( h(x) )

The function ( h(x)=\log x - 8 ) is a vertical shift of the parent function ( f(x)=\log x ) down by 8 units. Vertical shifts do not affect the vertical asymptote, domain, or the general shape related to the x - values.

Step3: Determine Asymptote

For the parent function ( y = \log x ), the vertical asymptote is ( x = 0 ). Since vertical shifts do not change the vertical asymptote, the vertical asymptote of ( h(x)=\log x - 8 ) is also ( x = 0 ).

Step4: Determine Domain

The domain of a logarithmic function ( y=\log_b(x) ) (here ( b = 10 ) for common logarithm) is the set of all positive real numbers. The vertical shift does not change the values of ( x ) for which the function is defined. So, the domain of ( h(x)=\log x - 8 ) is ( (0, \infty) ) (all real numbers greater than 0).

Step5: Determine Range

The range of the parent function ( y = \log x ) is ( (-\infty, \infty) ). A vertical shift down by 8 units still results in a range of all real numbers. So, the range of ( h(x)=\log x - 8 ) is ( (-\infty, \infty) ).

Answer:

• Asymptote: ( x = 0 )
• Domain: ( (0, \infty) )
• Range: ( (-\infty, \infty) ) (For graphing: The graph of ( h(x)=\log x - 8 ) is the graph of ( y = \log x ) shifted down 8 units. The vertical asymptote ( x = 0 ) is drawn as a dashed line.)