Unit 2 - Functions
Required Prior Knowledge
Questions
a) Substitute \(a=3\), \(b=2\) and \(c=16\) into \(a^{2}-b+\sqrt{c}\)
b) Expand and simplify the expression \(2\left(x+2\right)^{2}-3\left(x+2\right)\)
c) Solve the equation \(x^{2}+5x-14=0\)
Solutions
Get Ready
Questions
Write the following using algebra:
a) Double a number then add seven
b) Add seven to a number then double
c) Square a number then divide by 3
d) Divide a number by 3 then square it
e) Subtract two from a number, then square root
f) Square root a number then subtract two
Solutions
Notes
Since a function has a unique value from the range for each possible value in the domain, we can think of it as a machine that takes an input value and gives a output value based on the input.
The function “machine” is given by the name \(f\). We write this algebraically in one of two ways:
\(f\left(x\right)=2x+3\) which is read as “\(f\) of \(x\) is equal to \(2x+3\)”
\(f:x\mapsto 2x+3\) which is read as “\(f\) maps \(x\) to \(2x+3\)”
Using this notation \(f\left(2\right)\) which is read as “\(f\) of \(2\)” means we are inputting the value \(2\) into the function \(f\). Thus$$f\left(2\right)=2\left(2\right)+3=7$$This means that the point \(\left(2,7\right)\) is on the graph of the function \(y=f\left(x\right)\).
Examples and Your Turns
Example
Let \(f\left(x\right)=3x+7\). Find \(f\left(2\right)\).
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$$\begin{align}f\left(2\right)&=3\left(2\right)+7\\&=13\end{align}$$
Your Turn
Let \(f\left(x\right)=3x+7\). Find \(f\left(5\right)\).
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$$\begin{align}f\left(5\right)&=3\left(5\right)+7\\&=22\end{align}$$
Your Turn
Let \(g\left(x\right)=x^{2}-5x+1\). Find \(g\left(-2\right)\).
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$$\begin{align}g\left(-2\right)&=\left(-2\right)^{2}-5\left(-2\right)+1\\&=15\end{align}$$
Your Turn
Let \(f\left(x\right)=\frac{2-3x}{x^{2}+1}\). Find \(f\left(-3\right)\).
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$$\begin{align}f\left(-3\right)&=\frac{2-3\left(-3\right)}{\left(-3\right)^{2}+1}\\&=\frac{11}{10}\end{align}$$
Example
Let \(g\left(x\right)=5x-9\). Solve \(g\left(x\right)=21\).
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$$\begin{align}g\left(x\right)&=21\\ 5x-9&=21\\5x&=30\\x&=6\end{align}$$
Your Turn
Let \(g\left(x\right)=5x-9\). Solve \(g\left(x\right)=-64\).
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$$\begin{align}g\left(x\right)&=-64\\ 5x-9&=-64\\5x&=-55\\x&=-11\end{align}$$
Your Turn
Let \(h\left(x\right)=\frac{3x-2}{x}\). Solve \(h\left(x\right)=-5\).
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$$\begin{align}h\left(x\right)&=-5\\ \frac{3x-2}{x}&=-5\\3x-2&=-5x\\8x&=2\\x&=\frac{1}{4}\end{align}$$
Your Turn
Let \(h\left(x\right)=\frac{23-x}{x+1}\). Solve \(h\left(x\right)=11\).
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$$\begin{align}h\left(x\right)&=11\\ \frac{23-x}{x+1}&=11\\23-x&=11x+11\\12&=12x\\x&=1\end{align}$$
Your Turn
Let \(f\left(x\right)=x^{2}-3x\). Solve \(f\left(x\right)=4\).
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$$\begin{align}f\left(x\right)&=4\\ x^{2}-3x&=4\\x^{2}-3x-4&=0\\ \left(x+1\right)\left(x-4\right)&=0\\x=-1, x=4\end{align}$$
Your Turn
Let \(f\left(x\right)=x^{2}+7x\). Solve \(f\left(x\right)=8\).
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$$\begin{align}f\left(x\right)&=8\\ x^{2}+7x&=8\\x^{2}+7x-8&=0\\ \left(x+8\right)\left(x-1\right)&=0\\x=-8, x=1\end{align}$$
Example
Let \(f\left(x\right)=2x+1\). What is:
a) \(f\left(2x\right)\)
b) \(2f\left(x\right)\)
c) \(f\left(x^{2}\right)\)
d) \(f\left(x+1\right)\)
e) \(f\left(x\right)+1\)
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a) \(\begin{align}f\left(2x\right)&=2\left(2x\right)+1\\&=4x+1\end{align}\)
b) \(\begin{align}2f\left(x\right)&=2\left(2x+1\right)\\&=4x+2\end{align}\)
c) \(\begin{align}f\left(x^{2}\right)&=2\left(x^{2}\right)+1\\&=2x^{2}+1\end{align}\)
d) \(\begin{align}f\left(x+1\right)&=2\left(x+1\right)+1\\&=2x+3\end{align}\)
e) \(\begin{align}f\left(x\right)+1&=\left(2x+1\right)+1\\&=2x+2\end{align}\)
Your Turn
Let \(f\left(x\right)=3x-2\). What is:
a) \(f\left(4x\right)\)
b) \(4f\left(x\right)\)
c) \(f\left(x^{2}\right)\)
d) \(f\left(x-2\right)\)
e) \(f\left(x\right)-2\)
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a) \(\begin{align}f\left(4x\right)&=3\left(4x\right)-2\\&=12x-2\end{align}\)
b) \(\begin{align}4f\left(x\right)&=4\left(3x-2\right)\\&=12x-8\end{align}\)
c) \(\begin{align}f\left(x^{2}\right)&=3\left(x^{2}\right)-2\\&=3x^{2}-2\end{align}\)
d) \(\begin{align}f\left(x-2\right)&=3\left(x-2\right)-2\\&=3x-8\end{align}\)
e) \(\begin{align}f\left(x\right)-2&=\left(3x-2\right)-2\\&=3x-4\end{align}\)
Your Turn
Let \(f\left(x\right)=x^{2}+7\). What is:
a) \(f\left(3x\right)\)
b) \(3f\left(x\right)\)
c) \(f\left(x^{3}\right)\)
d) \(f\left(x+3\right)\)
e) \(f\left(x\right)+3\)
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a) \(\begin{align}f\left(3x\right)&=\left(3x\right)^{2}+7\\&=9x^{2}+7\end{align}\)
b) \(\begin{align}3f\left(x\right)&=3\left(x^{2}+7\right)\\&=3x^{2}+21\end{align}\)
c) \(\begin{align}f\left(x^{3}\right)&=\left(x^{3}\right)^{2}+7\\&=x^{6}+7\end{align}\)
d) \(\begin{align}f\left(x+3\right)&=\left(x+3\right)^{2}+7\\&=x^{2}+6x+16\end{align}\)
e) \(\begin{align}f\left(x\right)+3&=\left(x^{2}+7\right)+3\\&=x^{2}+10\end{align}\)
Your Turn
Let \(f\left(x\right)=x^{2}-3x\). What is:
a) \(f\left(\frac{1}{2}x\right)\)
b) \(\frac{1}{2}f\left(x\right)\)
c) \(f\left(\sqrt{x}\right)\)
d) \(f\left(x-\frac{1}{2}\right)\)
e) \(f\left(x\right)-\frac{1}{2}\)
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a) \(\begin{align}f\left(\frac{1}{2}x\right)&=\left(\frac{1}{2}x\right)^{2}-3\left(\frac{1}{2}x\right)\\&=\frac{1}{4}x^{2}-\frac{3}{2}x\end{align}\)
b) \(\begin{align}\frac{1}{2}f\left(x\right)&=\frac{1}{2}\left(x^{2}-3x\right)\\&=\frac{1}{2}x^{2}-\frac{3}{2}x\end{align}\)
c) \(\begin{align}f\left(\sqrt{x}\right)&=\left(\sqrt{x}\right)^{2}-3\left(\sqrt{x}\right)\\&=x-3\sqrt{x}\end{align}\)
d) \(\begin{align}f\left(x-\frac{1}{2}\right)&=\left(x-\frac{1}{2}\right)^{2}-3\left(x-\frac{1}{2}\right)\\&=x^{2}-x+\frac{1}{4}-3x+\frac{3}{2}\\&=x^{2}-4x+\frac{7}{4}\end{align}\)
e) \(\begin{align}f\left(x\right)-\frac{1}{2}&=\left(x^{2}-3x\right)-\frac{1}{2}\\&=x^{2}-3x-\frac{1}{2}\end{align}\)
Your Turn
Given that \(f\left(x\right)=2x-3\) and \(f\left(3a+1\right)=10\), calculate the value of \(a\).
Your Turn
Given that \(g\left(x\right)=x^{2}+1\) and \(g\left(2k-1\right)=6\), calculate the possible values of \(k\).
Your Turn
Given that \(f\left(x\right)=\frac{1}{x}\) and \(f\left(b+2\right)=\frac{1}{3}\), calculate the value of \(b\).
Your Turn
Given that \(h\left(x\right)=x^{2}-2x\) and \(h\left(a-1\right)=3\), calculate the possible values of \(a\).
Your Turn
Given that \(f\left(x\right)=x^{2}+3x-1\) and \(f\left(k+2\right)=1\), calculate the possible values of \(k\).
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
Explain, in your own words, what \(f\left(a\right)\) means.
Why is it important to use brackets when substituting into a function? Especially with negative numbers or algebraic expressions?
Common Mistakes / Misconceptions
Thinking that \(f\left(x\right)\) means \(f\times x\).
Misinterpreting \(f\left(x\right)=2\) to mean the input is \(2\).
Not substituting into ALL instances of the variable, for example in \(f\left(x\right)=x^{2}+2x\).
Errors when substituting negative values, especially with powers.
Connecting This to Other Skills
This particular skill is fundamentally essential for the development and mastery of all other function-related skills presented in Unit 2.
Specifically, this skill will be utilized extensively in the tasks of graphing functions (2.4), exploring composite functions (2.12), understanding inverse functions (2.13) and function transformations (2.18).
Additionally, function notation is further developed and built upon in the realm of differential calculus in unit 4, where students will explore more complex concepts and applications.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?