Topic 3.2 Functions and Sketching Graphs

Syllabus

Functions

3.2.1 determine when the composition of two functions is defined

3.2.2 determine the composition of two functions

3.2.3 determine if a function is one-to-one

3.2.4 find the inverse function of a one-to-one function

3.2.5 examine the reflection property of the graphs of a function and its inverse

Sketching graphs

3.2.6 use and apply |x| for the absolute value of the real number x and the graph of y = |x|

3.2.7 examine the relationship between the graph of y= f(x) and the graphs of y=frac1{f(x)} , y= |f(x)| and y= f(|x|)

3.2.8 sketch the graphs of simple rational functions where the numerator and denominator are polynomials of low degree

Lessons

1. Functions and their inverses

Note that the graphs in this Prezi are animated. Click on the play button to run the animation.
The wikipedia page on Set-builder notation is worth looking at for further details of how to specify domain and range.

2. Composite functions

3. Absolute values and functions

Graphs of absolute value functions in the general form aleft|b(x-c)right|+d
Generic absolute value graph

(click on the picture for an interactive version of the absolute value graph)

4. Graphs of (y=f(x)) and (y=frac1{f(x)})

5. Rational Functions

More about absolute value functions

It’s difficult to know just how much to include about absolute value functions. The information below is old content that may be inferred to be included by the broad curriculum descriptions.

Absolute value and inequalities:

To solve inequalities involving absolute values we make use of the relation (|a|leq b iff -bleq aleq b). For example:
begin{align*}<br /> |x-3| leq 9 &iff -9 le x-3 le 9 \<br /> &iff -6 le x le 12<br /> end{align*}

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