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Wize High School Grade 11 Math Textbook > Introduction to 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)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 + 2
where xx 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 tt.

distance(t)=2∣t−3∣\text{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)^2 , evaluate the following

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}


2. 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}

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Example 3

Give that f(x)f(x)is represented by the following graph, evaluate the following.

1. f(−2)f(-2)
f(−2)=1f(-2)=1

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


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

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

Practice: Function Notation

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

1. h(4)h(4) =

2. h(−3)h(-3) =




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

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

Practice: Function Notation


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


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

Practice: Function Notation


A pizzeria orders dough, costing the owner a flat fee of $25 for shipping, and $3.50 per kilogram of dough.

a) Write a function C(d)C\left(d\right) for the total cost of an order of dd kg of dough.
b) How much does it cost to order 10 kg of dough? How can we find this graphically?
c) If orders must come in a whole number of kilograms, how does this change the graph from part b)? What is the most dough the owner can purchase with $100?