Unit 2 - Functions
Required Prior Knowledge
Questions
For any general function, explain how you find the
a) \(y\)-intercept
b) roots
c) vertical asymptote
d) horizontal asymptote
Solutions
Get Ready
Questions
List as many different types of functions as you can (we have looked at 4, but there are more!).
Write down the details about them you can remember.
Solutions
Notes
A radical function is one of the form \(f\left(x\right)=\sqrt{ax+b}\).
The radicand (the function inside the radical) must be positive.
The natural domain of a radical function is \(\left\{x:ax+b\ge 0\right\}\).
The range of a radical function is \(\left\{y:y\ge 0\right\}\).
Examples and Your Turns
Example
Find the domain and range of the function$$y=\sqrt{x-3}$$
Your Turn
Find the domain and range of the function$$y=\sqrt{x+2}$$
-
Domain = \(\left\{x:x\ge -2\right\}\)
Range = \(\left\{y:y\ge 0\right\}\)
Your Turn
Find the domain and range of the function$$y=\sqrt{3-x}$$
-
Domain = \(\left\{x:x\le 3\right\}\)
Range = \(\left\{y:y\ge 0\right\}\)
Your Turn
Find the domain and range of the function$$y=\sqrt{4-2x}$$
-
Domain = \(\left\{x:x\le 2\right\}\)
Range = \(\left\{y:y\ge 0\right\}\)
Notes
Functions can be defined by different algebraic rules for different parts of their domain.
Such functions are called piecewise functions
Examples and Your Turns
Example
Consider the function$$f\left(x\right)=\begin{cases}-\left(x-2\right)^{2}, & \text{if }x<3 \\ x-4, & \text{if }x\ge 3 \end{cases}$$
a) Find \(f\left(0\right)\), \(f\left(3\right)\) and \(f\left(4\right)\).
b) Sketch \(y=f\left(x\right)\).
c) Write down the domain and range of \(f\).
Your Turn
Consider the function$$f\left(x\right)=\begin{cases}-\left(x^{2}-1\right), & \text{if }x\le 0 \\ 2, & \text{if }0<x<4 \\ 2+\sqrt{x}, & \text{if }x\ge 4 \end{cases}$$
a) Find \(f\left(-1\right)\), \(f\left(1\right)\), \(f\left(4\right)\) and \(f\left(9\right)\).
b) Sketch \(y=f\left(x\right)\).
c) Write down the domain and range of \(f\).
Notes
The absolute value or modulus of a real number \(x\) is the positive value of the number. It is the distance of the number along the number line from \(0\).
It is written as \(\left|x\right|\).
Some examples include \(\left|-7\right|=7\) or \(\left|5\right|=5\) or \(\left|-2.1\right|=2.1\) or \(\left|-\frac{1}{3}\right|=\frac{1}{3}\) or \(\left|3.7\right|=3.77\).
More formally, the absolute value function can be defined piecewise as$$\left|x\right|=\begin{cases}-x, & \text{if }x<0 \\ x, & \text{if }x\ge 0 \end{cases}$$The graph of \(y=\left|x\right|\) is shown below.
The natural domain of \(f\left(x\right)=\left|x\right|\) is \(\mathbb{R}\).
The range of \(f\left(x\right)=\left|x\right|\) is \(\left\{y:y\ge 0\right\}\).
We return to look at modulus functions when we look at graph transformations in Skill 2-18.
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
Why must the domain of a radical function be when the function inside the square root is greater than or equal to 0?
What are some real world examples of piecewise functions?
Why do we use the same notation for the absolute value of a number as the magnitude of a vector (Unit 9)and the modulus of a complex number (Unit 10)?
Common Mistakes / Misconceptions
Misreading the different domains for the functions within a piecewise function.
Connecting This to Other Skills
We will see radical functions again in calculus, specifically the Chain Rule (4.7).
Piecewise functions are useful for discussing concepts of limits and continuity (4.2).
The absolute value function is a form of function transformation (2.18).
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?