34. determine the domain and range of the exponential function, $f(x) = 3^x + 1$ shown in the graph. the…

34. 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. ${y: y in mathbb{r}}$

34. 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. ${y: y in mathbb{r}}$

Answer

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} )