Unit 2 - Functions
Required Prior Knowledge
Questions
For the function \(f\left(x\right)=x^{2}-3x+1\)
a) Evaluate \(f\left(2\right) \)
b) Solve \(f\left(x\right)=-1\)
c) Simplify \(f\left(2x-1\right)\)
Solutions
Get Ready
Questions
What are the domain and range of the following two functions?
a) \(f\left(x\right)=3x^{2}\)
b) \(g\left(x\right)=\sqrt{x-2}\)
Solutions
Notes
The composite function \(f\circ g\left(x\right)\) is the function obtained by applying function \(f\) to the output of function \(g\). That is$$f\circ g\left(x\right)=f\left(g\left(x\right)\right)$$We can represent this on a function machine like the one below.
Note that \(f\circ g\left(x\right)\) means we apply the function \(g\) first, then \(f\). We always apply the function closest to the \(x\) first.
The domain and range of the composite function might not be the same as the domain of the first function and the range of the second function. They interact in a way to create a new domain and range.
Examples and Your Turns
Example
Let \(f\left(x\right)=2x\) and \(g\left(x\right)=x^{2}+1\).
Find the domain and range of \(f\circ g\left(x\right)\).
Your Turn
Let \(f\left(x\right)=\frac{1}{x}\) and \(g\left(x\right)=x+1\).
Find the domain and range of \(f\circ g\left(x\right)\).
Your Turn
Let \(f\left(x\right)=\sqrt{x-10}\) and \(g\left(x\right)=x^{2}\).
Find the domain and range of \(f\circ g\left(x\right)\).
Your Turn
Let \(f\left(x\right)=\frac{1}{x-2}\) and \(g\left(x\right)=\sqrt{x}\).
Find the domain and range of \(f\circ g\left(x\right)\).
Example
Let \(f\left(x\right)=3x^{2}\) and \(g\left(x\right)=\sqrt{x-2}\).
Find:
(a) \(f\left(g\left(3\right)\right)\)
(b) \(f\circ g\left(x\right)\)
(c) the domain and range of \(f\circ g\left(x\right)\).
Your Turn
Let \(f\left(x\right)=3x^{2}\) and \(g\left(x\right)=\sqrt{x-2}\).
Find:
(a) \(g\left(f\left(3\right)\right)\)
(b) \(g\circ f\left(x\right)\)
(c) the domain and range of \(g\circ f\left(x\right)\).
Notes
From the last example and your turn, we can see that, in general$$f\circ g\left(x\right)\ne g\circ f\left(x\right)$$That is, composing functions is NOT commutative, and the order in which we compose the functions is very important as they will given different results.
Your Turn
Let \(f\left(x\right)=\frac{1}{x+1}\) and \(g\left(x\right)=2x+3\).
Find:
(a) \(f\circ g\left(x\right)\)
(b) \(g\circ f\left(x\right)\)
(c) the domain and range of \(f\circ g\left(x\right)\).
Your Turn
Let \(f\left(x\right)=\frac{x}{x-1}\) and \(g\left(x\right)=\frac{1}{x}\).
Find:
(a) \(f\circ g\left(x\right)\)
(b) \(g\circ f\left(x\right)\)
(c) the domain and range of \(f\circ g\left(x\right)\).
Your Turn
Consider two functions \(f\left(x\right)\) and \(g\left(x\right)\).
We know that \(g\left(x\right)=x+3\) and \(f\circ g\left(x\right)=4x^{2}+24x+38\).
Determine the expression for \(f\left(x\right)\).
Your Turn
Consider the functions \(f\left(x\right)=2x+a\) and \(g\left(x\right)=3x-5\), where \(a\) is a constant.
Find the value of \(a\) such that \(f\left(g\left(x\right)\right)=g\left(f\left(x\right)\right)\).
Your Turn
Consider two linear functions \(f\left(x\right)=ax+3\) and \(g\left(x\right)=2x+b\), where \(a,b\in\mathbb{R}\).
Given that \(f\left(g\left(x\right)\right)=g\left(f\left(x\right)\right)\) and \(f\circ g\left(1\right)=11\), determine the values of \(a\) and \(b\).
Your Turn
Let $$f\left(x\right)=\frac{x+1}{x+2}$$
(a) Find \(f\circ f\left(x\right)\)
(b) Find \(f\circ f\circ f\left(x\right)\)
(c) Keep going. What do you notice?
Investigation
Investigate what happens when you compose a combination of even and odd functions.
Challenge
Find a pair of functions such that $$f\circ g\left(5\right)=g\circ f\left(5\right)$$Find a pair of functions for which $$f\circ g\left(x\right)=g\circ f\left(x\right)$$for all \(x\).
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 is the order of functions important in a composite function? Can you think of a real world analogy where the order of actions changes the outcome?
Explain why the domain of the inner function is crucial to determining the domain of the composite function.
Explain why the domain of the outer function must be a subset of the range of the inner function.
Consider \(f\left(x\right)=x^{2}\) and \(g\left(x\right)=\sqrt{x}\). Are \(f\circ g\left(x\right)\) and \(g\circ f\left(x\right)\) the same? Why or why not? Are their domains the same?
Common Mistakes / Misconceptions
The most common misconception is the use the wrong order for \(f\circ g \left(x\right)\) and apply \(f\) first. The thought is that it is the first one we read, but it is NOT the first one applied to \(x\).
It is also common to confuse \(f\circ g\left(x\right)\) with \(f\left(x\right)g\left(x\right)\), and multiply the functions instead of finding the composite function.
Many students do not realise that the composite function \(f\circ g\) is itself a function, and can be treated as one.
If algebraic skills are not strong, it is easy to make algebraic errors in the manipulation of composite functions.
Connecting This to Other Skills
An understanding of what functions are (2.2) and domains and ranges of functions (2.6) is essential to understand composite functions.
The next skill, inverse functions (2.13) builds upon the idea of composite functions in a specific case.
When we get to calculus, we will encounter the Chain Rule (4.8) which uses composite functions. This is the single most common occurring skill in exams.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?