graph piecewise function
5+(1/(x-1) ) if x<1
f(x) 2x-4 if 1 ≤ x< 5
-1-x^2 if x ≥ 5
To graph a piecewise function like the one given, you need to break it down into its individual parts and plot them separately. Here's how you can do it:
1. Start by identifying the different conditions for the function and their corresponding expressions.
- For x < 1, the expression is f(x) = 5 + (1/(x-1)).
- For 1 ≤ x < 5, the expression is f(x) = 2x - 4.
- For x ≥ 5, the expression is f(x) = -1 - x^2.
2. Plotting the first condition: x < 1.
- Since the expression in this range involves a fraction, we need to consider the behavior as x approaches 1 from both sides.
- Take a few values slightly less than 1 (e.g., 0.5, 0.8, 0.9) and compute their corresponding y-values.
- Plot these points on the graph.
- Keep in mind that the function will have vertical asymptotes at x = 1, so the graph will approach those without actually crossing.
3. Plotting the second condition: 1 ≤ x < 5.
- Choose some values within this range (e.g., 2, 3, 4) and calculate their corresponding y-values.
- Plot these points on the graph as well.
4. Plotting the third condition: x ≥ 5.
- Select a few values greater than or equal to 5 (e.g., 5, 6, 7) and find their corresponding y-values.
- Plot these points on the graph.
5. Connect the plotted points on the graph to form a smooth curve.
- Note that there may be jumps or sharp changes in the graph at the transition points between different conditions.
By following these steps, you should be able to graph the given piecewise function.