Let's graph the exponential function [tex]\( g(x) = 4^{x-3} \)[/tex]. We'll begin by finding two points on the graph and identifying the asymptote. Finally, we'll determine the domain and range of the function.
### Step 1: Determine Two Points on the Graph
To find two points on the graph of [tex]\( g(x) \)[/tex], we can choose specific values for [tex]\( x \)[/tex] and calculate the corresponding [tex]\( y \)[/tex]-values from the function [tex]\( g(x) = 4^{x-3} \)[/tex].
#### Point 1: [tex]\( x = 2 \)[/tex]
[tex]\[ g(2) = 4^{2-3} = 4^{-1} = \frac{1}{4} = 0.25 \][/tex]
So, the first point is [tex]\( (2, 0.25) \)[/tex].
#### Point 2: [tex]\( x = 4 \)[/tex]
[tex]\[ g(4) = 4^{4-3} = 4^1 = 4 \][/tex]
So, the second point is [tex]\( (4, 4) \)[/tex].
### Step 2: Determine the Asymptote
For the function [tex]\( g(x) = 4^{x-3} \)[/tex], as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex], the value of [tex]\( 4^{x-3} \)[/tex] approaches 0. Therefore, the horizontal asymptote is [tex]\( y = 0 \)[/tex].
### Step 3: Identify the Domain
The domain of [tex]\( g(x) = 4^{x-3} \)[/tex] is all real numbers because exponential functions are defined for all real [tex]\( x \)[/tex]. Therefore, the domain is:
[tex]\[ \text{Domain: } (-\infty, \infty) \][/tex]
### Step 4: Identify the Range
For the range of [tex]\( g(x) = 4^{x-3} \)[/tex], since exponential functions of the form [tex]\( a^{x} \)[/tex] (where [tex]\( a > 1 \)[/tex]) are always positive, the range will be all positive real numbers. Therefore, the range is:
[tex]\[ \text{Range: } (0, \infty) \][/tex]
### Summary
To plot the exponential function [tex]\( g(x) = 4^{x-3} \)[/tex]:
1. Points on the graph: The points [tex]\( (2, 0.25) \)[/tex] and [tex]\( (4, 4) \)[/tex].
2. Asymptote: The horizontal asymptote at [tex]\( y = 0 \)[/tex].
3. Domain: [tex]\( (-\infty, \infty) \)[/tex].
4. Range: [tex]\( (0, \infty) \)[/tex].
You can now plot these points and draw the asymptote on your graph.
