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 x^{2} + a = 0. We have three cases:
- a > 0. In this case, the equation x^{2} + a = 0 has no real solutions and thus, the domain of f is the collection of all real numbers.
- a = 0. In this case, the equation x^{2} + a = 0 has only one solution x = 0. Thus, the domain of f is the collection of all non-zero real numbers.
- a < 0. In this case, the equation x^{2} + a = 0 has two solutions x = +a^{1/2}. Thus, the domain of f is the collection of all real numbers except +a^{1/2}.
- 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:
- a > 0. In this case, the range is the interval (0, 1/a].
- a = 0. In this case, the range is all the positive real numbers.
- a < 0. In this case, the range is the collection of all k > 0 and all k <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/(x^{3} + a)
- f(x) = 1/(x^{4} + a)
- f(x) = 1/(x^{5} + 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.]