Package of problems

Problem

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.

1. If $-5<a<b<-3$ then $f(a) < f(b).$

2. If $2.5 < a < b < 5$ then $f(a) < f(b).$

3. If $-2.5 < a< b <2$ then $f(a) > f(b).$

4. 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.

1. If $a$ is less than $b$, then $a^{2}$ is less than $b^{2}.$

2. If $a$ is less than $b$, then $a-5$ is less than $b-5.$

3. If $a$ is less than $b$, then $3-2a$ is less than $3-2b.$

4. 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 $b$?
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