Question

discuss what you notice about the domain, range, intercepts, and asymptote of $y=2^{x}+3$.
domain: $(-\infty,\infty)$
range: $(-5,\infty)$
asymptote:

1. graph y

Explanation:

1. Domain: The function $y=2^x + 3$ is an exponential function. Exponential functions accept all real numbers as input, so there are no restrictions on $x$.
2. Range: The parent function $2^x$ has a range of $(0, \infty)$. Adding 3 shifts the entire function upward by 3 units, so the lower bound of the range increases by 3.
3. Intercepts:

• x-intercept: Set $y=0$ and solve for $x$. $0=2^x + 3$ rearranges to $2^x=-3$, which has no real solution since exponential functions are always positive.
• y-intercept: Set $x=0$ and solve for $y$. $y=2^0 + 3 = 1 + 3 = 4$.

1. Asymptote: The parent function $2^x$ has a horizontal asymptote at $y=0$. Shifting the function up 3 units moves this asymptote up by 3 units.

Answer:

• Domain: $(-\infty, \infty)$
• Range: $(3, \infty)$
• Intercepts: y-intercept at $(0, 4)$; no real x-intercept
• Asymptote: Horizontal asymptote at $y=3$