Sure, let's analyze the given exponential function [tex]\( y = 3^x + 2 \)[/tex] step by step to determine its [tex]$y$[/tex]-intercept, asymptote, domain, and range.
### 1. Finding the y-intercept:
The [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex]. To find this, we substitute [tex]\( x = 0 \)[/tex] into the function and solve for [tex]\( y \)[/tex]:
[tex]\[ y = 3^0 + 2 \][/tex]
Since [tex]\( 3^0 = 1 \)[/tex]:
[tex]\[ y = 1 + 2 \][/tex]
[tex]\[ y = 3 \][/tex]
So, the [tex]\( y \)[/tex]-intercept is [tex]\((0, 3)\)[/tex].
### 2. Finding the Asymptote:
For the given function [tex]\( y = 3^x + 2 \)[/tex], we observe the behavior of the function as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex]. The term [tex]\( 3^x \)[/tex] approaches 0 because any number raised to a large negative power tends to 0.
[tex]\[ y \approx 0 + 2 \][/tex]
Thus, as [tex]\( x \)[/tex] goes to [tex]\(-\infty\)[/tex], [tex]\( y \)[/tex] approaches 2. Therefore, the function has a horizontal asymptote at:
[tex]\[ y = 2 \][/tex]
### 3. Finding the Domain:
The domain of an exponential function is the set of all possible values of [tex]\( x \)[/tex]. Since there are no restrictions on the values that [tex]\( x \)[/tex] can take for the function [tex]\( y = 3^x + 2 \)[/tex]:
[tex]\[ \text{Domain} = \text{all real numbers} \][/tex]
or in interval notation:
[tex]\[ (-\infty, \infty) \][/tex]
### 4. Finding the Range:
The range of the function is the set of all possible values of [tex]\( y \)[/tex]. To find the range, we analyze the output values of [tex]\( y = 3^x + 2 \)[/tex]:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( 3^x \to 0 \)[/tex], so [tex]\( y \to 2 \)[/tex].
- As [tex]\( x \to \infty \)[/tex], [tex]\( 3^x \to \infty \)[/tex], so [tex]\( y \to \infty \)[/tex].
Since [tex]\( y \)[/tex] can get arbitrarily close to 2 but never actually reaches 2 and continues to increase without bound, the range is:
[tex]\[ (2, \infty) \][/tex]
### Summary:
- y-intercept: [tex]\((0, 3)\)[/tex]
- Asymptote: [tex]\( y = 2 \)[/tex]
- Domain: all real numbers ([tex]\( -\infty, \infty \)[/tex])
- Range: [tex]\( (2, \infty) \)[/tex]
These are the key characteristics of the function [tex]\( y = 3^x + 2 \)[/tex].
