3. consider the function $y = 2^{-x}+2$.\n3a find the $y$-value of the $y$-intercept of the curve $y =…

3. consider the function $y = 2^{-x}+2$.\n3a find the $y$-value of the $y$-intercept of the curve $y = 2^{-x}+2$.\n$y = 3$\n3b find the equation of the horizontal asymptote of the curve $y = 2^{-x}+2$.\nenter your next step here

3. consider the function $y = 2^{-x}+2$.\n3a find the $y$-value of the $y$-intercept of the curve $y = 2^{-x}+2$.\n$y = 3$\n3b find the equation of the horizontal asymptote of the curve $y = 2^{-x}+2$.\nenter your next step here

Answer

Explanation:

Step1: Recall the concept of horizontal asymptote

For an exponential - type function of the form $y = a\cdot b^{mx}+k$, when $|b|\gt0$ and $b\neq1$, as $x\to\pm\infty$, we analyze the behavior of the exponential part. In the function $y = 2^{-x}+2=\left(\frac{1}{2}\right)^{x}+2$, as $x\to+\infty$, the term $\left(\frac{1}{2}\right)^{x}\to0$.

Step2: Determine the horizontal asymptote

Since $\lim_{x\to+\infty}\left(\frac{1}{2}\right)^{x}=0$, then $\lim_{x\to+\infty}(2^{-x}+2)=0 + 2$. The equation of the horizontal asymptote is $y = 2$.

Answer:

$y = 2$