use transformations of the graph of (f(x)=3^{x}) to graph the given function. be sure to graph and give the…

use transformations of the graph of (f(x)=3^{x}) to graph the given function. be sure to graph and give the equation of the asymptote. use the graph to determine the functions domain and range. if applicable, use a graphing utility to confirm your hand - drawn graphs.\ng(x)=3^{x}-6

use transformations of the graph of (f(x)=3^{x}) to graph the given function. be sure to graph and give the equation of the asymptote. use the graph to determine the functions domain and range. if applicable, use a graphing utility to confirm your hand - drawn graphs.\ng(x)=3^{x}-6

Answer

Explanation:

Step1: Recall the properties of $y = 3^x$

The function $y = 3^x$ has a horizontal - asymptote at $y = 0$, domain $(-\infty,\infty)$ and range $(0,\infty)$.

Step2: Analyze the transformation of $g(x)=3^x - 6$

The function $g(x)=3^x - 6$ is a vertical shift of the function $y = 3^x$ down by 6 units.

Step3: Find the asymptote

For the function $g(x)=3^x - 6$, the horizontal asymptote is $y=-6$ because the entire graph of $y = 3^x$ is shifted down 6 units.

Step4: Determine the domain

The domain of an exponential function of the form $y = a^x + k$ (where $a>0,a\neq1$ and $k$ is a real - number) is the set of all real numbers. So, the domain of $g(x)=3^x - 6$ is $(-\infty,\infty)$.

Step5: Determine the range

Since the range of $y = 3^x$ is $(0,\infty)$ and we shift it down 6 units, the range of $g(x)$ is $(-6,\infty)$.

Answer:

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