Problem:
Plot the graphs of the functions

for values of a between -4 and 4. Find the domain and range of f.

Visualization:
In the following LiveMath Notebook, you can start and stop the animation by clicking on the graph. The value of a is indicated in the upper right-hand corner of the graph.

• Domain of f: The domain of f consists of all real numbers except where x2 + a = 0. We have three cases:
1. a > 0. In this case, the equation x2 + a = 0 has no real solutions and thus, the domain of f is the collection of all real numbers.
2. a = 0. In this case, the equation x2 + a = 0 has only one solution x = 0. Thus, the domain of f is the collection of all non-zero real numbers.
3. a < 0. In this case, the equation x2 + a = 0 has two solutions x = +a1/2. Thus, the domain of f is the collection of all real numbers except +a1/2.

• Range of f: The range of f consists of all real numbers k where we can solve the equation

Solving this equation, we get

First, we note that k = 0 is never in the range of f. Next, we note that 1/k must be greater than or equal to a. Again, we have three cases:

1. a > 0. In this case, the range is the interval (0, 1/a].
2. a = 0. In this case, the range is all the positive real numbers.
3. a < 0. In this case, the range is the collection of all k > 0 and all <1/a.

You can look at the graphs of some of these cases by changing the value of a in the notebook above.

You can change the function in the notebook above. Try

• f(x) = 1/(x3 + a)
• f(x) = 1/(x4 + a)
• f(x) = 1/(x5 + a)

[To change the definition of f(x) in the notebook above, highlight the exponent in right-hand side of the equation and then type in the new exponent.]