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Sagot :
To graph the exponential function [tex]\( g(x) = 3^x + 3 \)[/tex], let's follow these steps:
1. Plotting Points:
To graph the function, we should start by plotting a couple of points [tex]\( (x, g(x)) \)[/tex]. Let's calculate the values for [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex]:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 3^0 + 3 = 1 + 3 = 4 \][/tex]
So, the point is [tex]\( (0, 4) \)[/tex].
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = 3^1 + 3 = 3 + 3 = 6 \][/tex]
So, the point is [tex]\( (1, 6) \)[/tex].
We can plot these points: [tex]\( (0, 4) \)[/tex] and [tex]\( (1, 6) \)[/tex].
2. Asymptote:
The function [tex]\( g(x) = 3^x + 3 \)[/tex] has a horizontal asymptote. As [tex]\( x \)[/tex] goes to negative infinity, [tex]\( 3^x \)[/tex] approaches zero. Hence, the horizontal asymptote is [tex]\( y = 3 \)[/tex]. This is a horizontal line that the graph approaches but never actually touches.
3. Graphing:
With the points [tex]\( (0, 4) \)[/tex] and [tex]\( (1, 6) \)[/tex] plotted, and knowing that the horizontal asymptote is [tex]\( y = 3 \)[/tex], we can sketch the curve.
- The graph will rise steeply as [tex]\( x \)[/tex] increases because [tex]\( 3^x \)[/tex] grows exponentially.
- As [tex]\( x \)[/tex] decreases, the graph will approach the horizontal line [tex]\( y = 3 \)[/tex] without ever touching it.
4. Domain and Range:
- The domain of [tex]\( g(x) \)[/tex], which are the allowable [tex]\( x \)[/tex]-values, is all real numbers. Hence:
[tex]\[ \text{Domain}: (-\infty, \infty) \][/tex]
- The range of [tex]\( g(x) \)[/tex] is the set of [tex]\( y \)[/tex]-values that the function can take. Since [tex]\( 3^x \)[/tex] is always positive and as [tex]\( x \to -\infty \)[/tex], [tex]\( 3^x \)[/tex] approaches 0, and thus [tex]\( g(x) = 3^x + 3 \)[/tex] always stays above 3. Therefore:
[tex]\[ \text{Range}: (3, \infty) \][/tex]
So, here is a summary of the steps:
1. Plot the points [tex]\( (0, 4) \)[/tex] and [tex]\( (1, 6) \)[/tex].
2. Draw the horizontal asymptote at [tex]\( y = 3 \)[/tex].
3. Sketch the curve passing through the points and approaching the asymptote.
Finally, ensure to graph the function accurately reflecting its exponential nature with the key points and asymptote.
### Corrected Range
Based on the function definition:
[tex]\[ \text{Range}: (3, \infty) \][/tex]
Final Notes: If your initial input on the range was incorrect, please update to [tex]\( (3, \infty) \)[/tex].
1. Plotting Points:
To graph the function, we should start by plotting a couple of points [tex]\( (x, g(x)) \)[/tex]. Let's calculate the values for [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex]:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 3^0 + 3 = 1 + 3 = 4 \][/tex]
So, the point is [tex]\( (0, 4) \)[/tex].
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = 3^1 + 3 = 3 + 3 = 6 \][/tex]
So, the point is [tex]\( (1, 6) \)[/tex].
We can plot these points: [tex]\( (0, 4) \)[/tex] and [tex]\( (1, 6) \)[/tex].
2. Asymptote:
The function [tex]\( g(x) = 3^x + 3 \)[/tex] has a horizontal asymptote. As [tex]\( x \)[/tex] goes to negative infinity, [tex]\( 3^x \)[/tex] approaches zero. Hence, the horizontal asymptote is [tex]\( y = 3 \)[/tex]. This is a horizontal line that the graph approaches but never actually touches.
3. Graphing:
With the points [tex]\( (0, 4) \)[/tex] and [tex]\( (1, 6) \)[/tex] plotted, and knowing that the horizontal asymptote is [tex]\( y = 3 \)[/tex], we can sketch the curve.
- The graph will rise steeply as [tex]\( x \)[/tex] increases because [tex]\( 3^x \)[/tex] grows exponentially.
- As [tex]\( x \)[/tex] decreases, the graph will approach the horizontal line [tex]\( y = 3 \)[/tex] without ever touching it.
4. Domain and Range:
- The domain of [tex]\( g(x) \)[/tex], which are the allowable [tex]\( x \)[/tex]-values, is all real numbers. Hence:
[tex]\[ \text{Domain}: (-\infty, \infty) \][/tex]
- The range of [tex]\( g(x) \)[/tex] is the set of [tex]\( y \)[/tex]-values that the function can take. Since [tex]\( 3^x \)[/tex] is always positive and as [tex]\( x \to -\infty \)[/tex], [tex]\( 3^x \)[/tex] approaches 0, and thus [tex]\( g(x) = 3^x + 3 \)[/tex] always stays above 3. Therefore:
[tex]\[ \text{Range}: (3, \infty) \][/tex]
So, here is a summary of the steps:
1. Plot the points [tex]\( (0, 4) \)[/tex] and [tex]\( (1, 6) \)[/tex].
2. Draw the horizontal asymptote at [tex]\( y = 3 \)[/tex].
3. Sketch the curve passing through the points and approaching the asymptote.
Finally, ensure to graph the function accurately reflecting its exponential nature with the key points and asymptote.
### Corrected Range
Based on the function definition:
[tex]\[ \text{Range}: (3, \infty) \][/tex]
Final Notes: If your initial input on the range was incorrect, please update to [tex]\( (3, \infty) \)[/tex].
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