Question

1. determine the domain and range of the exponential function, $f(x) = 3^x + 1$ shown in the graph. the domain is... a. ${x: x < 0}$ b. ${x: x > 1}$ c. ${x: x > 2}$ d. ${x: x in mathbb{r}}$ the range is... a. ${y: y < 0}$ b. ${y: y > 1}$ c. ${y: y > 2}$ d. ${y: y in mathbb{r}}$

Explanation:

Step1: Analyze Domain of \( f(x) = 3^x + 1 \)

The function \( 3^x \) is an exponential function, and the domain of any exponential function \( a^x \) (where \( a>0, a
eq1 \)) is all real numbers (\( \mathbb{R} \)). Adding 1 to \( 3^x \) does not restrict the input values (x - values). So the domain of \( f(x)=3^x + 1 \) is all real numbers, i.e., \( \{x: x\in\mathbb{R}\} \), which corresponds to option D for the domain.

Step2: Analyze Range of \( f(x) = 3^x + 1 \)

The range of \( 3^x \) is \( \{y: y > 0\} \) because exponential functions with base \( a>1 \) (here \( a = 3 \)) have outputs greater than 0. When we add 1 to \( 3^x \), we shift the graph up by 1 unit. So the range of \( 3^x+1 \) becomes \( \{y: y > 0 + 1\} = \{y: y > 1\} \), which corresponds to option B for the range.

Answer:

Domain: D. \( \{x: x\in\mathbb{R}\} \)
Range: B. \( \{y: y > 1\} \)