To graph the exponential function [tex]\( g(x) = 3^x - 2 \)[/tex]:
1. Understand the Function's Behavior:
- The function [tex]\( g(x) = 3^x - 2 \)[/tex] is an exponential function shifted downward by 2 units.
- Horizontal Asymptote: The horizontal asymptote is determined by observing the behavior as [tex]\( x \to -\infty \)[/tex]. For [tex]\( g(x) \)[/tex], the function approaches [tex]\(-2\)[/tex], so the horizontal asymptote is [tex]\( y = -2 \)[/tex].
- Domain: The domain of [tex]\( g(x) \)[/tex] is all real numbers, written as [tex]\( (-\infty, \infty) \)[/tex].
- Range: Since an exponential function [tex]\( 3^x \)[/tex] always yields positive values and is shifted downwards by 2 units, the range is [tex]\( [-2, \infty) \)[/tex].
2. Choose Points to Plot:
- Point 1: When [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = 3^0 - 2 = 1 - 2 = -1
\][/tex]
So, the point at [tex]\( x=0 \)[/tex] is (0, -1).
- Point 2: When [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = 3^1 - 2 = 3 - 2 = 1
\][/tex]
So, the point at [tex]\( x=1 \)[/tex] is (1, 1).
3. Graphing the Function:
- Draw the horizontal asymptote at [tex]\( y = -2 \)[/tex]. This helps guide the shape of the graph.
- Plot the points (0, -1) and (1, 1).
- Sketch the curve:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( g(x) \)[/tex] approaches [tex]\(-2\)[/tex] but never quite reaches it, hugging the horizontal asymptote.
- As [tex]\( x \to \infty \)[/tex], [tex]\( g(x) \)[/tex] increases exponentially without bound.
4. Graph Appearance:
- The graph will look like a standard exponential curve shifted down by 2 units.
- The key features are the points (0, -1) and (1, 1) and the horizontal asymptote [tex]\( y = -2 \)[/tex].
5. Conclusion on Domain and Range:
- Domain: The function [tex]\( g(x) = 3^x - 2 \)[/tex] is defined for all real values of [tex]\( x \)[/tex], so the domain is [tex]\( (-\infty, \infty) \)[/tex].
- Range: The lowest value the function can get close to is [tex]\(-2\)[/tex], but as [tex]\( x \)[/tex] increases, the function increases without bound. Therefore, the range is [tex]\( [-2, \infty) \)[/tex].
By following these steps, you can graph the function [tex]\( g(x) = 3^x - 2 \)[/tex] and identify its key characteristics, including its domain and range.
