Alright, let's graph the function [tex]\( g(x) = 3^x - 3 \)[/tex] step by step, identify key points and the asymptote, and then determine the domain and range of the function.
### Step-by-Step Solution:
1. Understanding the Function:
The function [tex]\( g(x) = 3^x - 3 \)[/tex] is a transformed exponential function where the term [tex]\( 3^x \)[/tex] is shifted down by 3 units.
2. Plotting Key Points:
We will calculate the values of [tex]\( g(x) \)[/tex] for a few specific [tex]\( x \)[/tex]-values:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = 3^0 - 3 = 1 - 3 = -2
\][/tex]
Point: [tex]\( (0, -2) \)[/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = 3^1 - 3 = 3 - 3 = 0
\][/tex]
Point: [tex]\( (1, 0) \)[/tex]
- When [tex]\( x = -1 \)[/tex]:
[tex]\[
g(-1) = 3^{-1} - 3 = \frac{1}{3} - 3 = \frac{1}{3} - \frac{9}{3} = -\frac{8}{3}
\][/tex]
Point: [tex]\( (-1, -\frac{8}{3}) \approx (-1, -2.67) \)[/tex]
- When [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) = 3^2 - 3 = 9 - 3 = 6
\][/tex]
Point: [tex]\( (2, 6) \)[/tex]
3. Identify the Asymptote:
Since [tex]\( 3^x \)[/tex] has a horizontal asymptote at [tex]\( y = 0 \)[/tex], [tex]\( g(x) = 3^x - 3 \)[/tex] will have a horizontal asymptote shifted down by 3 units. Therefore, the horizontal asymptote of [tex]\( g(x) \)[/tex] is:
[tex]\[
y = -3
\][/tex]
4. Sketching the Graph:
On a coordinate plane:
- Plot the points [tex]\( (0, -2) \)[/tex], [tex]\( (1, 0) \)[/tex], [tex]\( (-1, -2.67) \approx (-1, -8/3) \)[/tex], and [tex]\( (2, 6) \)[/tex].
- Draw a smooth curve that passes through these points.
- Ensure the curve approaches the horizontal asymptote [tex]\( y = -3 \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex] but never touches or crosses it.
The graph will rise rapidly as [tex]\( x \)[/tex] increases and will approach [tex]\( y = -3 \)[/tex] but never touch it as [tex]\( x \)[/tex] decreases.
### Domain and Range:
Domain:
The exponential function [tex]\( 3^x \)[/tex] is defined for all real numbers, and the transformation does not restrict this. Therefore, the domain of [tex]\( g(x) = 3^x - 3 \)[/tex] is:
[tex]\[
\text{Domain: } (-\infty, \infty)
\][/tex]
Range:
As [tex]\( x \)[/tex] increases, [tex]\( 3^x \)[/tex] becomes very large, meaning [tex]\( 3^x - 3 \)[/tex] also becomes very large. As [tex]\( x \)[/tex] decreases, [tex]\( 3^x \)[/tex] approaches 0 from the positive side, meaning [tex]\( 3^x - 3 \)[/tex] approaches [tex]\(-3\)[/tex] from above but never reaches [tex]\(-3\)[/tex]. Therefore, the range of [tex]\( g(x) = 3^x - 3 \)[/tex] is:
[tex]\[
\text{Range: } (-3, \infty)
\][/tex]
### Summary:
- Domain: [tex]\( (-\infty, \infty) \)[/tex]
- Range: [tex]\( (-3, \infty) \)[/tex]
