Wize High School Algebra I Textbook (Common Core) > Relations & Functions

Function Notation

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Function Notation
Function Notation
When writing the equation or rule for a function, we can use function notation.

Function notation shows us the
Name of the function
Input variable
Equation or rule that connects the independent and dependent variables  


When reading function notation f(x)f(x)f(x)we say "f of x" or "f at x." 

Example 1

A cab company charges riders $2 for a ride, plus $1.50 for every mile traveled.  
Write an equation that represents the cost of a cab ride using function notation.

Cost(x)=1.5x+2Cost(x)=1.5x + 2Cost(x)=1.5x+2
where xxx is the number of miles.

Wize Tip
Some common names for functions include the letters, f, g, and h, but we can call functions a variety of names.
This following function is called "distance," and has an input variable of ttt.

distance(t)=2∣t−3∣\text{distance}(t) = 2|t-3|distance(t)=2∣t−3∣

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Substitution using function notation
If the input variable is replaced by another value, then we can substitute into the equation or graph.  

Evaluating this will give us the output of the function.

Example 2

Given that g(x)=−(x+1)2g(x)=-(x+1)^2g(x)=−(x+1)2 , evaluate the following

1.  g(1)g(1)g(1)

g(1)=−(1+1)2=−(2)2=−4\begin{aligned}
g(1) &= -(1+1)^2 \\
&= -(2)^2 \\
&= -4
\end{aligned}g(1)​=−(1+1)2=−(2)2=−4​

2. g(3a)g(3a)g(3a)

g(3a)=−(3a+1)2=−(9a2+6a+1)=−9a2−6a−1\begin{aligned}
g(3a) &= -(3a + 1)^2 \\
&=-(9a^2 + 6a + 1) \\
&=-9a^2 - 6a - 1
\end{aligned}g(3a)​=−(3a+1)2=−(9a2+6a+1)=−9a2−6a−1​

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Example 3

Give that f(x)f(x)f(x)is represented by the following graph, evaluate the following.
  
1.  f(−2)f(-2)f(−2)

  
f(−2)=1f(-2)=1f(−2)=1

2.  f(0)f(0)f(0)

  
f(0)=1f(0)=1f(0)=1

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Example Function Notation

At a carnival game you are asked to pick a number, add 5 to it, square the result, and then subtract this value from 15.

a.  Use function notation to describe this process.

f(x)=15−(x+5)2f(x)=15-(x+5)^2f(x)=15−(x+5)2

b.  What number do you end up with if you start with -1,2, or 0?

f(−1)=−1f(-1) = -1f(−1)=−1
f(2)=−34f(2)=-34f(2)=−34
f(0)=−10f(0)=-10f(0)=−10

Practice: Function Notation
Let h(x)=5x+7h(x) = 5x + 7h(x)=5x+7.  Evaluate the following:

1. h(4)h(4)h(4) = 

2. h(−3)h(-3)h(−3) = 

Let g(x)=2x2−x+3g(x) = 2x^2 - x + 3g(x)=2x2−x+3.  Evaluate the following:

1.  g(0)g(0)g(0) = 
2.  g(4)g(4)g(4) =

I don't know

Practice: Function Notation

The graph of the function f(x)=12(x−1)2−2f(x)=\frac{1}{2}(x-1)^2-2f(x)=21​(x−1)2−2 is given.

What input is needed so that f(x)=−2f(x) = -2f(x)=−2 ?