# 3.1 Functions and function notation  (Page 9/21)

 Page 9 / 21

We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown in [link] .

Toolkit Functions
Name Function Graph
Constant $f\left(x\right)=c,\text{\hspace{0.17em}}$ where $c$ is a constant
Identity $f\left(x\right)=x$
Absolute value $f\left(x\right)=|x|$
Quadratic $f\left(x\right)={x}^{2}$
Cubic $f\left(x\right)={x}^{3}$
Reciprocal $f\left(x\right)=\frac{1}{x}$
Reciprocal squared $f\left(x\right)=\frac{1}{{x}^{2}}$
Square root $f\left(x\right)=\sqrt{x}$
Cube root $f\left(x\right)=\sqrt[3]{x}$

Access the following online resources for additional instruction and practice with functions.

## Key equations

 Constant function $f\left(x\right)=c,$ where $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ is a constant Identity function $f\left(x\right)=x$ Absolute value function $f\left(x\right)=|x|$ Quadratic function $f\left(x\right)={x}^{2}$ Cubic function $f\left(x\right)={x}^{3}$ Reciprocal function $f\left(x\right)=\frac{1}{x}$ Reciprocal squared function $f\left(x\right)=\frac{1}{{x}^{2}}$ Square root function $f\left(x\right)=\sqrt{x}$ Cube root function $f\left(x\right)=\sqrt[3]{x}$

## Key concepts

• A relation is a set of ordered pairs. A function is a specific type of relation in which each domain value, or input, leads to exactly one range value, or output. See [link] and [link] .
• Function notation is a shorthand method for relating the input to the output in the form $\text{\hspace{0.17em}}y=f\left(x\right).\text{\hspace{0.17em}}$ See [link] and [link] .
• In tabular form, a function can be represented by rows or columns that relate to input and output values. See [link] .
• To evaluate a function, we determine an output value for a corresponding input value. Algebraic forms of a function can be evaluated by replacing the input variable with a given value. See [link] and [link] .
• To solve for a specific function value, we determine the input values that yield the specific output value. See [link] .
• An algebraic form of a function can be written from an equation. See [link] and [link] .
• Input and output values of a function can be identified from a table. See [link] .
• Relating input values to output values on a graph is another way to evaluate a function. See [link] .
• A function is one-to-one if each output value corresponds to only one input value. See [link] .
• A graph represents a function if any vertical line drawn on the graph intersects the graph at no more than one point. See [link] .
• The graph of a one-to-one function passes the horizontal line test. See [link] .

## Verbal

What is the difference between a relation and a function?

A relation is a set of ordered pairs. A function is a special kind of relation in which no two ordered pairs have the same first coordinate.

What is the difference between the input and the output of a function?

Why does the vertical line test tell us whether the graph of a relation represents a function?

When a vertical line intersects the graph of a relation more than once, that indicates that for that input there is more than one output. At any particular input value, there can be only one output if the relation is to be a function.

How can you determine if a relation is a one-to-one function?

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nothing up todat yet
Miranda
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jai
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jai
Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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jai
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Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
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Jeffrey
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Miranda
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Jeffrey
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Miranda
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Jeffrey
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Miranda
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Miranda
Jeffrey
Jeffrey
Miranda
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Miranda
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Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
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Miranda
what is the solution of the given equation?
which equation
Miranda
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Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
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Ye
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Nokwanda
C'est comment
Abdel
Hi
Amanda
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SORIE
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Chinni
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Chinni
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Hassan
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Abdel
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Yaona
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function