use transformations of the graph of ( f(x)=e^{x} ) to graph the given function. be sure to give the equation…

use transformations of the graph of ( f(x)=e^{x} ) to graph the given function. be sure to give the equation of the asymptote. use the graph to determine the functions domain and range.\n( h(x)=e^{3 x}-6 )\ngraph ( h(x) ) and its asymptote. graph the asymptote as a dashed line. use the graphing tool to graph the function.

use transformations of the graph of ( f(x)=e^{x} ) to graph the given function. be sure to give the equation of the asymptote. use the graph to determine the functions domain and range.\n( h(x)=e^{3 x}-6 )\ngraph ( h(x) ) and its asymptote. graph the asymptote as a dashed line. use the graphing tool to graph the function.

Answer

Explanation:

Step1: Recall the parent function ( y = e^x )

The parent function ( y=e^x ) has a horizontal asymptote ( y = 0 ), domain ( (-\infty,\infty) ), and range ( (0,\infty) ).

Step2: Analyze the transformation for ( h(x)=e^{3x}-6 )

For the function ( h(x)=e^{3x}-6 ), we consider the general form of an exponential function ( y = a\cdot e^{bx}+c ). Here ( a = 1 ), ( b=3 ), ( c=-6 ). The horizontal asymptote of ( y = e^{3x}-6 ) is found by considering the limit as ( x\to-\infty ). Since ( \lim_{x\to-\infty}e^{3x}=0 ), the horizontal asymptote is ( y=-6 ). The domain of an exponential function ( y = e^{u(x)}+k ) (where ( u(x)=3x ) in our case) is all real numbers because the exponential function ( e^{3x} ) is defined for all ( x\in R ). So, the domain of ( h(x) ) is ( (-\infty,\infty) ). For the range, we know that ( e^{3x}>0 ) for all ( x\in R ). Then ( e^{3x}-6>-6 ). So the range is ( (-6,\infty) ).

Answer:

Asymptote: ( y = - 6 ), Domain: ( (-\infty,\infty) ), Range: ( (-6,\infty) )