Graph
f(x)= {x+1 x<0}
f(x)= {- sqrt x x>=0}
To graph the function f(x) = {x+1 x<0, - sqrt x x>=0}, we need to graph the two distinct parts of the function separately and indicate the points where the function transitions from one part to the other.
First, let's graph the function f(x) = x + 1 for x < 0.
Since it is a linear function, the graph is a straight line with a slope of 1 and a y-intercept of 1. This part of the graph will lie entirely above the x-axis for x < 0.
Next, let's graph the function f(x) = - sqrt(x) for x ≥ 0.
Since it is a square root function, the graph will start at the origin (0,0) and move downwards along the positive x-axis. This part of the graph will lie entirely below the x-axis for x ≥ 0.
Now, let's combine the two parts of the graph.
At x = 0, there is a transition point. The graph will abruptly 'jump' from the linear function to the square root function at this point.
Here is the graph of the function f(x) = {x+1 x<0, - sqrt x x>=0}:
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Note: The graph above is not drawn to scale and is only a rough approximation to illustrate the behavior of the function.
To graph the function f(x) = x + 1 for x < 0 and f(x) = -√x for x ≥ 0, we will follow these steps:
Step 1: Find the x-intercept(s)
To find the x-intercept, we set f(x) equal to 0 and solve for x:
x + 1 = 0
x = -1
Step 2: Plot the x-intercept(s)
Plot the x-intercept (-1, 0) on the graph.
Step 3: Determine the behavior for x < 0
For x < 0, the function is f(x) = x + 1.
Pick a few x-values less than 0, such as -3, -2, and -1. Substitute these values into the equation to find the corresponding y-values:
f(-3) = -3 + 1 = -2
f(-2) = -2 + 1 = -1
f(-1) = -1 + 1 = 0
Step 4: Plot the points for x < 0
Plot the points (-3, -2), (-2, -1), and (-1, 0) on the graph.
Step 5: Determine the behavior for x ≥ 0
For x ≥ 0, the function is f(x) = -√x.
Pick a few x-values greater than or equal to 0, such as 0, 1, and 4. Substitute these values into the equation to find the corresponding y-values:
f(0) = -√0 = 0
f(1) = -√1 = -1
f(4) = -√4 = -2
Step 6: Plot the points for x ≥ 0
Plot the points (0, 0), (1, -1), and (4, -2) on the graph.
Step 7: Connect the dots
Draw a smooth curve connecting all the plotted points, taking care to reflect the different behavior for x < 0 and x ≥ 0.
Final Result:
The graph of the function f(x) = x + 1 for x < 0 and f(x) = -√x for x ≥ 0 should resemble a diagonal line passing through the points (-3, -2), (-2, -1), and (-1, 0), and a downward curve passing through the points (0, 0), (1, -1), and (4, -2).
To graph the function f(x) = {x+1 x<0; -√x x≥0}, we can follow these steps:
Step 1: Identify the domain and range of the function.
Domain: All real numbers.
Range: For x < 0, the range will be all real numbers. For x ≥ 0, the range will be all non-positive real numbers.
Step 2: Plot the points for x < 0.
For x < 0, the function is defined as f(x) = x + 1. You can choose some values for x less than 0, such as -3, -2, -1, and 0, and calculate the corresponding y values using the equation f(x) = x + 1.
For example:
For x = -3: f(-3) = -3 + 1 = -2, so we have the point (-3, -2).
For x = -2: f(-2) = -2 + 1 = -1, so we have the point (-2, -1).
For x = -1: f(-1) = -1 + 1 = 0, so we have the point (-1, 0).
Step 3: Plot the point for x ≥ 0 and draw the graph.
For x≥ 0, the function is defined as f(x) = -√x. Choose some values for x greater than or equal to 0, such as 0, 1, 4, and 9, and calculate the corresponding y values using the equation f(x) = -√x.
For example:
For x = 0: f(0) = -√0 = 0, so we have the point (0, 0).
For x = 1: f(1) = -√1 = -1, so we have the point (1, -1).
For x = 4: f(4) = -√4 = -2, so we have the point (4, -2).
For x = 9: f(9) = -√9 = -3, so we have the point (9, -3).
Now, let's plot all the points on the coordinate plane and connect the dots for the different regions of the function. For x < 0, draw a straight line connecting (-3, -2), (-2, -1), and (-1, 0). For x ≥ 0, plot the points (0, 0), (1, -1), (4, -2), (9, -3), and connect them smoothly with a curve passing through all of them.
The resulting graph will consist of a straight line for x < 0 and a curve for x ≥ 0.