what are the domain, range, and asymptote of h(x) = 6^x - 4?\no domain: {x | x is a real number}; range: {y…

Explanation:

Step1: Determine the domain

For an exponential - function of the form $y = a^x$ (where $a>0,a\neq1$), the domain of $y = 6^x$ is all real numbers. The function $h(x)=6^x - 4$ is a transformation of the exponential function $y = 6^x$. Since there are no restrictions on the value of $x$ for which the function $h(x)$ is undefined, the domain of $h(x)$ is ${x|x\text{ is a real number}}$.

Step2: Analyze the range of $y = 6^x$

The exponential function $y = 6^x$ has a range of ${y|y>0}$ because for any real - number $x$, $6^x>0$.

Step3: Determine the range of $h(x)=6^x - 4$

Let $y = h(x)=6^x - 4$. Since $6^x>0$, then $y=6^x - 4>-4$. So the range of $h(x)$ is ${y|y > - 4}$.

Step4: Find the asymptote

The exponential function $y = 6^x$ has a horizontal asymptote at $y = 0$. For the function $h(x)=6^x - 4$, which is a vertical translation of $y = 6^x$ down 4 units, the horizontal asymptote is $y=-4$.

Answer:

domain: ${x|x\text{ is a real number}}$; range: ${y|y > - 4}$; asymptote: $y=-4$