Graph each function;identify the domain and range;and compare the graph with the graph of y=1/x.
1.y=-10/x
Domain:
Range:
Compare:
2.y=1/(x+3)+3
Domain:
Range:
Compare:
1. Graph of y = -10/x:
To graph this function, we can start by making a table of x and y values. Choose some x values, calculate the corresponding y values, and plot them on a graph.
For example, let's choose x = -5, -2, 0, 2, and 5:
When x = -5, y = -10/(-5) = 2
When x = -2, y = -10/(-2) = 5
When x = 0, y = -10/0 = undefined (as division by zero is not defined)
When x = 2, y = -10/2 = -5
When x = 5, y = -10/5 = -2
Plotting these points on a graph, we get a graph that goes through the points (-5, 2), (-2, 5), (2, -5), and (5, -2). The graph approaches but never reaches the x-axis and the y-axis.
Domain: The domain of the function y = -10/x is all real numbers except for x = 0, as division by zero is undefined.
Range: The range of the function y = -10/x is all real numbers except for y = 0, since there is no x value that produces a y value of 0.
Comparison with y = 1/x: The graph of y = -10/x is similar to the graph of y = 1/x, but it is vertically stretched by a factor of 10 and reflected in the x-axis.
2. Graph of y = 1/(x + 3) + 3:
To graph this function, we can follow a similar process as before.
Let's choose x = -5, -3, -2, 0, and 2:
When x = -5, y = 1/(-5 + 3) + 3 = 2
When x = -3, y = 1/(-3 + 3) + 3 = undefined (division by zero)
When x = -2, y = 1/(-2 + 3) + 3 = 4
When x = 0, y = 1/(0 + 3) + 3 = 4/3 or approximately 1.33
When x = 2, y = 1/(2 + 3) + 3 = 3.2
Plotting these points on a graph, we get a graph that passes through the points (-5, 2), (-2, 4), (0, 4/3), and (2, 3.2). The graph approaches but never reaches the x-axis and the vertical asymptote x = -3.
Domain: The domain of the function y = 1/(x + 3) + 3 is all real numbers except for x = -3, as division by zero is undefined.
Range: The range of the function y = 1/(x + 3) + 3 is all real numbers except for y = 3, since there is no x value that produces a y value of 3.
Comparison with y = 1/x: The graph of y = 1/(x + 3) + 3 is a horizontal shift of the graph of y = 1/x to the left by 3 units and a vertical shift upwards by 3 units. It also has a vertical asymptote at x = -3.