Describe the graphs of the functions f(x) = 2x – 3 and g(x) = –2x – 3. Compare and contrast the domain and range of f(x) and g(x).
both straight lines, first has a slope of 2 and y-intercept of -3
the 2nd has a slope of -2 and also a y-intercept of -3
both have domains and ranges of x ∈ R
To describe the graphs of the functions f(x) = 2x - 3 and g(x) = -2x - 3, we can start by examining their slopes and y-intercepts.
For f(x) = 2x - 3, the slope is 2, meaning that the graph will have a positive slope and will go upward as x increases. The y-intercept is -3, which is where the graph intersects the y-axis.
For g(x) = -2x - 3, the slope is -2, indicating that the graph has a negative slope and will go downward as x increases. The y-intercept is also -3.
Both graphs are straight lines, but they have different slopes. The graph of f(x) = 2x - 3 has a positive slope, while the graph of g(x) = -2x - 3 has a negative slope.
Now, let's compare and contrast the domain and range of f(x) and g(x).
The domain of a function refers to the set of all possible x-values. Since both functions are linear, their domains are all real numbers, so the domain for f(x) and g(x) is (-∞, ∞), which means that the functions are defined for all real values of x.
The range of a function, on the other hand, represents the set of all possible y-values. Since both functions are linear, their ranges are also all real numbers. However, the specific y-values that the graphs of f(x) and g(x) can reach will differ due to their different slopes. The range for f(x) will be all real numbers greater than or equal to the y-intercept (-3), as the graph goes upward. The range for g(x) will be all real numbers less than or equal to the y-intercept (-3), as the graph goes downward.
In summary, the graphs of f(x) = 2x - 3 and g(x) = -2x - 3 are both straight lines, but with different slopes. The domain of both functions is all real numbers, while the range of f(x) includes all real numbers greater than or equal to -3, and the range of g(x) includes all real numbers less than or equal to -3.
To describe the graphs of the functions f(x) = 2x - 3 and g(x) = -2x - 3, we can analyze their slopes and y-intercepts.
For f(x) = 2x - 3, the slope is 2, which means the graph is a straight line that is increasing. The y-intercept is -3, so the graph crosses the y-axis at the point (0, -3). From this point, we can plot additional points by moving 1 unit to the right and 2 units up, or moving 1 unit to the left and 2 units down. This gives us a graph that slants upwards as we move to the right, and slants downwards as we move to the left.
For g(x) = -2x - 3, the slope is -2, which means the graph is a straight line that is decreasing. The y-intercept is -3, so the graph also crosses the y-axis at the point (0, -3). From this point, we can plot additional points by moving 1 unit to the right and 2 units down, or moving 1 unit to the left and 2 units up. This gives us a graph that slants downwards as we move to the right, and slants upwards as we move to the left.
Comparing the graphs, we can see that they are mirror images of each other with respect to the x-axis. This is because the slopes have opposite signs, which determines the direction of the line, and the y-intercepts are equal.
Now let's discuss the domains and ranges of f(x) and g(x). The domain of a function represents all the possible x-values that the function can take. Since these are linear functions, the domain is the set of all real numbers, as there are no restrictions on x.
The range of a function represents all the possible y-values that the function can take. For f(x) = 2x - 3, as the graph slants upwards, we can see that the y-values increase infinitely. Therefore, the range of f(x) is also the set of all real numbers.
For g(x) = -2x - 3, as the graph slants downwards, we can see that the y-values decrease infinitely. Therefore, the range of g(x) is also the set of all real numbers.
In summary, the domains of both f(x) and g(x) are the set of all real numbers, and the ranges of both f(x) and g(x) are also the set of all real numbers.