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Sagot :
Sure! Let's graph the exponential function [tex]\( g(x) = -2^x - 3 \)[/tex], plot points, draw the curve, and determine the domain and range.
### Step-by-Step Solution:
1. Understand the Function: The function [tex]\( g(x) = -2^x - 3 \)[/tex] is an exponential function where the base is 2, but it is negated and shifted down by 3 units.
2. Plot Points:
- To graph the function accurately, we start by plotting a few points. We'll choose a couple of [tex]\( x \)[/tex] values.
For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = -2^0 - 3 = -1 - 3 = -4 \][/tex]
So, the point [tex]\((0, -4)\)[/tex] is on the graph.
For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = -2^1 - 3 = -2 - 3 = -5 \][/tex]
So, the point [tex]\((1, -5)\)[/tex] is on the graph.
3. Draw the Curve:
- Plot the points [tex]\((0, -4)\)[/tex] and [tex]\((1, -5)\)[/tex] on the coordinate plane.
- The function [tex]\( g(x) = -2^x - 3 \)[/tex] is an exponential decay function with a vertical shift. As [tex]\( x \)[/tex] increases, the value of [tex]\( -2^x \)[/tex] becomes more negative, and then we subtract 3.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 2^x \)[/tex] approaches 0, so [tex]\( g(x) \)[/tex] approaches [tex]\(-3\)[/tex].
- The curve will be approaching the horizontal asymptote at [tex]\( y = -3 \)[/tex] from below but will never touch it.
4. Domain and Range:
- Domain: Exponential functions are defined for all real numbers. Thus, the domain is:
[tex]\[ (-\infty, \infty) \][/tex]
- Range: Since [tex]\( g(x) = -2^x - 3 \)[/tex] will never reach [tex]\(-3\)[/tex] (the horizontal asymptote), and as [tex]\( x \)[/tex] decreases, [tex]\( g(x) \)[/tex] keeps going more negative. Thus, the range is:
[tex]\[ (-\infty, -3) \][/tex]
### Summary:
- The graph of [tex]\( g(x) = -2^x - 3 \)[/tex] passes through the points [tex]\((0, -4)\)[/tex] and [tex]\((1, -5)\)[/tex].
- The domain of the function is [tex]\( (-\infty, \infty) \)[/tex].
- The range of the function is [tex]\( (-\infty, -3) \)[/tex].
### Graph:
To graph this function properly, you can follow these steps and plot the points mentioned, then sketch the approaching horizontal asymptote and the curve.
I hope this helps! If you have any other questions, feel free to ask.
### Step-by-Step Solution:
1. Understand the Function: The function [tex]\( g(x) = -2^x - 3 \)[/tex] is an exponential function where the base is 2, but it is negated and shifted down by 3 units.
2. Plot Points:
- To graph the function accurately, we start by plotting a few points. We'll choose a couple of [tex]\( x \)[/tex] values.
For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = -2^0 - 3 = -1 - 3 = -4 \][/tex]
So, the point [tex]\((0, -4)\)[/tex] is on the graph.
For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = -2^1 - 3 = -2 - 3 = -5 \][/tex]
So, the point [tex]\((1, -5)\)[/tex] is on the graph.
3. Draw the Curve:
- Plot the points [tex]\((0, -4)\)[/tex] and [tex]\((1, -5)\)[/tex] on the coordinate plane.
- The function [tex]\( g(x) = -2^x - 3 \)[/tex] is an exponential decay function with a vertical shift. As [tex]\( x \)[/tex] increases, the value of [tex]\( -2^x \)[/tex] becomes more negative, and then we subtract 3.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 2^x \)[/tex] approaches 0, so [tex]\( g(x) \)[/tex] approaches [tex]\(-3\)[/tex].
- The curve will be approaching the horizontal asymptote at [tex]\( y = -3 \)[/tex] from below but will never touch it.
4. Domain and Range:
- Domain: Exponential functions are defined for all real numbers. Thus, the domain is:
[tex]\[ (-\infty, \infty) \][/tex]
- Range: Since [tex]\( g(x) = -2^x - 3 \)[/tex] will never reach [tex]\(-3\)[/tex] (the horizontal asymptote), and as [tex]\( x \)[/tex] decreases, [tex]\( g(x) \)[/tex] keeps going more negative. Thus, the range is:
[tex]\[ (-\infty, -3) \][/tex]
### Summary:
- The graph of [tex]\( g(x) = -2^x - 3 \)[/tex] passes through the points [tex]\((0, -4)\)[/tex] and [tex]\((1, -5)\)[/tex].
- The domain of the function is [tex]\( (-\infty, \infty) \)[/tex].
- The range of the function is [tex]\( (-\infty, -3) \)[/tex].
### Graph:
To graph this function properly, you can follow these steps and plot the points mentioned, then sketch the approaching horizontal asymptote and the curve.
I hope this helps! If you have any other questions, feel free to ask.
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