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 $y = a^{x}+b$ (in this case $h(x)=6^{x}-4$ where $a = 6$ and $b=-4$), the domain of an exponential function $y = a^{x}$ is all real numbers. Since there are no restrictions on the value of $x$ for which $h(x)=6^{x}-4$ is defined, the domain of $h(x)$ is ${x|x\text{ is a real number}}$.

Step2: Analyze the range and asymptote

The general form of an exponential function is $y = a^{x}$, and for $y = 6^{x}$, the range is ${y|y>0}$ because $6^{x}>0$ for all real - valued $x$. For the function $h(x)=6^{x}-4$, we shift the graph of $y = 6^{x}$ down by 4 units. So, if $y = 6^{x}>0$, then $h(x)=6^{x}-4>-4$. The horizontal asymptote of $y = 6^{x}$ is $y = 0$. When we shift the graph of $y = 6^{x}$ down by 4 units, the horizontal asymptote of $h(x)=6^{x}-4$ is $y=-4$.

Answer:

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