Certainly! Let's graph the function [tex]\( g(x) = 2^x - 3 \)[/tex] step-by-step, plot points, draw the asymptote, and determine the domain and range.
### Step-by-Step Solution
#### 1. Understanding the Function
The function [tex]\( g(x) = 2^x - 3 \)[/tex] is an exponential function that has been shifted vertically downward by 3 units. The base of the exponential function is 2, which means it grows exponentially as [tex]\( x \)[/tex] increases.
#### 2. Asymptote
For the function [tex]\( g(x) = 2^x - 3 \)[/tex]:
- As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]), [tex]\( 2^x \)[/tex] approaches 0.
- Therefore, [tex]\( g(x) \)[/tex] approaches [tex]\( -3 \)[/tex].
So, the horizontal asymptote of this function is [tex]\( y = -3 \)[/tex].
#### 3. Plotting Key Points
We need at least two points to plot on the graph for a clear understanding of the function’s behavior.
Choose [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = 2^{-1} - 3 = \frac{1}{2} - 3 = -2.5 \][/tex]
Choose [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = 2^2 - 3 = 4 - 3 = 1 \][/tex]
So, the points we will plot are:
- [tex]\( (-1, -2.5) \)[/tex]
- [tex]\( (2, 1) \)[/tex]
#### 4. Graphing the Function
- Plot the two points [tex]\((-1, -2.5)\)[/tex] and [tex]\( (2, 1) \)[/tex] on the coordinate plane.
- Draw a smooth curve through these points that gets closer and closer to the horizontal asymptote [tex]\( y = -3 \)[/tex] as [tex]\( x \)[/tex] goes to negative infinity.
- As [tex]\( x \)[/tex] increases, the graph will rise exponentially.
#### 5. Domain and Range
Domain: The domain of [tex]\( g(x) = 2^x - 3 \)[/tex] is all real numbers because the function is defined for every real number [tex]\( x \)[/tex].
[tex]\[ \text{Domain: } (-\infty, \infty) \][/tex]
Range: The range of [tex]\( g(x) = 2^x - 3 \)[/tex] is all real numbers greater than [tex]\( -3 \)[/tex]. The function never reaches [tex]\( -3 \)[/tex], but it gets arbitrarily close to it.
[tex]\[ \text{Range: } (-3, \infty) \][/tex]
### Summary
- Domain: [tex]\((- \infty, \infty)\)[/tex]
- Range: [tex]\((-3, \infty)\)[/tex]
By following the above steps, you can accurately plot the graph of the function [tex]\( g(x) = 2^x - 3 \)[/tex], identify key points, draw the asymptote, and determine its domain and range.
