Use transformations to graph the following function. Determine the domain, ra horizontal asymptote, and $$ y $$-intercept of the function. $$ f(x)=3^{x}+4 $$ The domain of $$ f(x) $$ is $$ (-\infty, \infty) $$. (Type your answer in interval notation.) The range of $$ f(x) $$ is $$ (4, \infty) $$. (Type your answer in interval notation.) Determine the horizontal asymptote(s), if any, of the function $$ f(x)=3^{x}+4 $$. Selec the correct choice below and, if necessary, fill in the answer box $$ ( $$ es) to complete your choice. and the bottom asymptote is (Type equations.) B. The function has one horizontal asymptote, $$ y=4 $$. (Type an equation.) C. The function has no horizontal asymptotes. The $$ y $$-intercept of the function is (Type an integer or a simplified fraction.)

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UpStudy AI · Accepted Answer

To analyze the function $$ f(x) = 3^x + 4 $$, we can break down the information step by step.

### Domain
The function $$ f(x) = 3^x + 4 $$ is defined for all real numbers since the exponential function $$ 3^x $$ is defined for all $$ x $$. Therefore, the domain is:
$$
\text{Domain: } (-\infty, \infty)
$$

### Range
The exponential function $$ 3^x $$ approaches $$ 0 $$ as $$ x $$ approaches $$ -\infty $$ and increases without bound as $$ x $$ approaches $$ \infty $$. Therefore, $$ 3^x + 4 $$ approaches $$ 4 $$ as $$ x $$ approaches $$ -\infty $$ and increases without bound as $$ x $$ approaches $$ \infty $$. Thus, the range is:
$$
\text{Range: } (4, \infty)
$$

### Horizontal Asymptote
As $$ x $$ approaches $$ -\infty $$, $$ f(x) $$ approaches $$ 4 $$. Therefore, the function has a horizontal asymptote at:
$$
y = 4
$$
So the correct choice is:
$$
\text{B. The function has one horizontal asymptote, } y = 4.
$$

### $$ y $$-Intercept
To find the $$ y $$-intercept, we evaluate $$ f(0) $$:
$$
f(0) = 3^0 + 4 = 1 + 4 = 5
$$
Thus, the $$ y $$-intercept is:
$$
\text{The } y\text{-intercept of the function is } 5.
$$

### Summary
- **Domain:** $$ (-\infty, \infty) $$
- **Range:** $$ (4, \infty) $$
- **Horizontal Asymptote:** $$ y = 4 $$
- **$$ y $$-Intercept:** $$ 5 $$
Domain: $$ (-\infty, \infty) $$ Range: $$ (4, \infty) $$ Horizontal Asymptote: $$ y = 4 $$ $$ y $$-Intercept: $$ 5 $$

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