Use the interactivity below to find values of \(a\) and \(b\) for which
\(a\) is less than \(b\) and \(f(a)\) is less than \(f(b).\)
\(a\) is less than \(b\) but \(f(a)\) is greater than \(f(b).\)
Here are some statements about the function \(f(x)\) in the graph above. In each case, decide if the statement is true. If it is not true, try to adapt it to make a true statement.
If \(-5<a<b<-3\) then \(f(a) < f(b).\)
If \(2.5 < a < b < 5\) then \(f(a) < f(b).\)
If \(-2.5 < a< b <2\) then \(f(a) > f(b).\)
If \(-2.5 < a < b < 3\) then \(f(a) > f(b).\)
Now consider the following statements about real numbers \(a\) and \(b\) and some familiar functions. Try to sketch or visualise the graph of the functions as you think about the statements.
If \(a\) is less than \(b\), then \(a^{2}\) is less than \(b^{2}.\)
If \(a\) is less than \(b\), then \(a-5\) is less than \(b-5.\)
If \(a\) is less than \(b\), then \(3-2a\) is less than \(3-2b.\)
If \(a\) is less than \(b\), then \(\dfrac{1}{a}\) is greater than \(\dfrac{1}{b}.\)
If you think a statement is true or false for all \(a\) and \(b\), can you justify this using a graph?
If a statement may be true for some values of \(a\) and \(b\) and false for other values, can you form a true statement or statements by restricting the set of values for \(a\) and \(b\)?
