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 with domain all real numbers (\( \mathbb{R} \)). Adding 1 (a vertical shift) does not change the domain. So the domain of \( f(x)=3^x + 1 \) is all real numbers, matching option D for domain.

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

The range of \( 3^x \) is \( y > 0 \) (since exponential functions \( a^x \) with \( a>1 \) have range \( y>0 \)). For \( f(x)=3^x + 1 \), we shift the graph up by 1 unit. So the range becomes \( y > 0 + 1 = 1 \), which matches option B for range.

Answer:

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