start with the graph of the appropriate basic exponential function f and use transformations to sketch the…
start with the graph of the appropriate basic exponential function f and use transformations to sketch the graph of the function g. state the domain and range of g and the horizontal asymptote of its graph.\ng(x)=2^{x - 2}+2\nuse the graphing tool to graph the function.
Answer
Explanation:
Step1: Identify the basic function
The basic exponential function is $y = 2^{x}$.
Step2: Analyze horizontal - shift
For the function $g(x)=2^{x - 2}+2$, the $x-2$ inside the exponent causes a horizontal shift of 2 units to the right compared to $y = 2^{x}$.
Step3: Analyze vertical - shift
The + 2 outside the exponent causes a vertical shift of 2 units up compared to $y = 2^{x - 2}$.
Step4: Find the domain
The domain of an exponential function of the form $y = a^{x - h}+k$ is all real numbers. So, the domain of $g(x)$ is $(-\infty,\infty)$.
Step5: Find the range
The range of the basic exponential function $y = 2^{x}$ is $(0,\infty)$. After the vertical - shift of 2 units up, the range of $g(x)$ is $(2,\infty)$.
Step6: Find the horizontal asymptote
The horizontal asymptote of the basic exponential function $y = 2^{x}$ is $y = 0$. After the vertical - shift of 2 units up, the horizontal asymptote of $g(x)$ is $y = 2$.
Answer:
Domain: $(-\infty,\infty)$; Range: $(2,\infty)$; Horizontal asymptote: $y = 2$