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Graph the exponential function [tex]g(x)=3^x-2[/tex].

1. Plot two points on the graph of the function.
2. Draw the asymptote.
3. Click on the graph-a-function button.

Additionally, give the domain and range of the function using interval notation:

- Domain: [tex](-\infty, \infty)[/tex]
- Range: [tex][-2, \infty)[/tex]

Sagot :

GinnyAnswer
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.
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