Answer :
Certainly! Let's graph the exponential function [tex]\( g(x) = 4^{x+3} \)[/tex]. We'll plot two points, draw the asymptote, and finally determine the domain and range. Here's a step-by-step solution:
### 1. Understanding the Function
The function [tex]\( g(x) = 4^{x+3} \)[/tex] is an exponential function with a base of 4. The general form of an exponential function is [tex]\( a^{(x+c)} \)[/tex], where [tex]\( a \)[/tex] is the base and [tex]\( c \)[/tex] is a constant.
### 2. Choosing Points to Plot
We can choose two points for visualization:
- Let’s choose [tex]\( x = -2 \)[/tex]
- Let’s choose [tex]\( x = 2 \)[/tex]
#### For [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = 4^{(-2+3)} = 4^1 = 4 \][/tex]
#### For [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = 4^{(2+3)} = 4^5 = 1024 \][/tex]
So, the points to plot are [tex]\((-2, 4)\)[/tex] and [tex]\((2, 1024)\)[/tex].
### 3. Drawing the Asymptote
The horizontal asymptote of an exponential function [tex]\( g(x) = 4^{x+3} \)[/tex] is [tex]\( y = 0 \)[/tex]. This is because, as [tex]\( x \)[/tex] approaches negative infinity, the exponential value approaches 0 but never actually reaches it.
### 4. Graphical Representation
#### (Manually or using graphing software, plot the following):
- The point [tex]\((-2, 4)\)[/tex]
- The point [tex]\((2, 1024)\)[/tex]
- Draw a smooth curve passing through these points showing the exponential growth.
- Draw the horizontal asymptote at [tex]\( y = 0 \)[/tex].
### 5. Domain and Range
#### Domain:
An exponential function [tex]\( g(x) = 4^{x+3} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. Therefore, the domain is:
[tex]\[ \text{Domain: } (-\infty, \infty) \][/tex]
#### Range:
Since [tex]\( 4^{x+3} \)[/tex] is always positive for any real number [tex]\( x \)[/tex] (it never reaches 0 or goes negative), the range is:
[tex]\[ \text{Range: } (0, \infty) \][/tex]
### Summary of the findings:
- Points: [tex]\((-2, 4)\)[/tex] and [tex]\((2, 1024)\)[/tex]
- Asymptote: [tex]\( y = 0 \)[/tex]
- Domain: [tex]\( (-\infty, \infty) \)[/tex]
- Range: [tex]\( (0, \infty) \)[/tex]
### Conclusion:
To correctly visualize and describe the function [tex]\( g(x)=4^{x+3} \)[/tex]:
1. Plot the points [tex]\((-2, 4)\)[/tex] and [tex]\((2, 1024)\)[/tex].
2. Draw the asymptote at [tex]\( y = 0 \)[/tex].
3. Clearly note that the function grows exponentially from left to right.
Ensure you use graphing tools appropriately if doing this manually or digitally.
### 1. Understanding the Function
The function [tex]\( g(x) = 4^{x+3} \)[/tex] is an exponential function with a base of 4. The general form of an exponential function is [tex]\( a^{(x+c)} \)[/tex], where [tex]\( a \)[/tex] is the base and [tex]\( c \)[/tex] is a constant.
### 2. Choosing Points to Plot
We can choose two points for visualization:
- Let’s choose [tex]\( x = -2 \)[/tex]
- Let’s choose [tex]\( x = 2 \)[/tex]
#### For [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = 4^{(-2+3)} = 4^1 = 4 \][/tex]
#### For [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = 4^{(2+3)} = 4^5 = 1024 \][/tex]
So, the points to plot are [tex]\((-2, 4)\)[/tex] and [tex]\((2, 1024)\)[/tex].
### 3. Drawing the Asymptote
The horizontal asymptote of an exponential function [tex]\( g(x) = 4^{x+3} \)[/tex] is [tex]\( y = 0 \)[/tex]. This is because, as [tex]\( x \)[/tex] approaches negative infinity, the exponential value approaches 0 but never actually reaches it.
### 4. Graphical Representation
#### (Manually or using graphing software, plot the following):
- The point [tex]\((-2, 4)\)[/tex]
- The point [tex]\((2, 1024)\)[/tex]
- Draw a smooth curve passing through these points showing the exponential growth.
- Draw the horizontal asymptote at [tex]\( y = 0 \)[/tex].
### 5. Domain and Range
#### Domain:
An exponential function [tex]\( g(x) = 4^{x+3} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. Therefore, the domain is:
[tex]\[ \text{Domain: } (-\infty, \infty) \][/tex]
#### Range:
Since [tex]\( 4^{x+3} \)[/tex] is always positive for any real number [tex]\( x \)[/tex] (it never reaches 0 or goes negative), the range is:
[tex]\[ \text{Range: } (0, \infty) \][/tex]
### Summary of the findings:
- Points: [tex]\((-2, 4)\)[/tex] and [tex]\((2, 1024)\)[/tex]
- Asymptote: [tex]\( y = 0 \)[/tex]
- Domain: [tex]\( (-\infty, \infty) \)[/tex]
- Range: [tex]\( (0, \infty) \)[/tex]
### Conclusion:
To correctly visualize and describe the function [tex]\( g(x)=4^{x+3} \)[/tex]:
1. Plot the points [tex]\((-2, 4)\)[/tex] and [tex]\((2, 1024)\)[/tex].
2. Draw the asymptote at [tex]\( y = 0 \)[/tex].
3. Clearly note that the function grows exponentially from left to right.
Ensure you use graphing tools appropriately if doing this manually or digitally.
