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Domain & Range

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     Domain
The domain of a function refers to the complete set of possible values of the independent variable. (i.e.: the domain is the set of all possible x-values which will make my function "work" and will out-put real y-values). 


Example 1
The function y=x2y=x^2y=x2 is a quadratic function. Let's look at its graph:


Considering all the possible values of 'x' that allow us to get a real number in return, we are allowed to take any real number 'x'.
Therefore, the domain of y=x2y=x^2y=x2 is all real numbers. 
Interval Notation: (−∞,∞)(-\infin,\infin)(−∞,∞)
Set Notation: {x∈R∣ x is any real number}\{x\in\mathbb{R}|~x~\text{is any real number\}}{x∈R∣ x is any real number}
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Example 2
Consider the graph of the function below:

Considering all the possible values of 'x' that allow us to get a real number in return, we are allowed to take any values of 'x' that are equal to or greater than 5. 

Therefore, the domain of the function is x≥5x\geq5x≥5
Interval Notation: [5,∞)[5,\infty)[5,∞)
Set Notation: {x∈R∣ x≥5}\{x\in\mathbb{R}|~x\geq{5}\}{x∈R∣ x≥5}

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Range
The range of a function refers to the complete set of possible resulting values of the dependent variable. (i.e.: the range is the resulting y-values we get after substituting all possible x values into our function).


Example 1
The function y=x2y=x^2y=x2 is a quadratic function. Let's look at its graph:

If we consider the possible return values for some input value, 'x', then we can see that 'y' is greater than or equal to 0.

Therefore, the range is y≥0y\geq0y≥0
Interval Notation: [0,∞)[0,\infin)[0,∞)
Set Notation: {y∈R∣ y≥0}\{y\in\mathbb{R}|~y\geq0\}{y∈R∣ y≥0}
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Example 2
Consider the graph of the function below:

If we consider the possible return values for some input value, 'x', then we can see that 'y' is greater than or equal to 5. 

Therefore, the range is y≥5y\geq{5}y≥5
Interval Notation: [5,∞)[5,\infin)[5,∞)
Set Notation: {y∈R∣ y≥5}\{y\in\mathbb{R}|~y\geq{5}\}{y∈R∣ y≥5}

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Example: Domain & Range
Find the domain & range of the following 

y=2x2−4y=2x^2-4y=2x2−4

1. Sketch the function:


2. Identify any maximums or minimums that could restrict the input and output values

xyMinimumNone-4MaximumNoneNone\begin{array}
{|c|c|c|}
\hline
&x&y\\\hline\\
\text{Minimum}&\text{None}&\text{-4}\\\\\hline
\\\text{Maximum}&\text{None}&\text{None}\\\\\hline
\end{array}MinimumMaximum​xNoneNone​y-4None​​


3. Express the domain & range in interval & set notation

DomainRangeInterval Notation(−∞,∞)[−4,∞)Set Notation{x∈R∣ x is any real number}{y∈R∣ y≥−4}\begin{array}
{|l|c|c|}\hline
\text{}&\text{Domain}&\text{Range}\\\hline\\

\text{Interval Notation}&(-\infin,\infin)&[-4,\infin)\\\\\hline\\

\text{Set Notation}&\{x\in\mathbb{R}|~x~\footnotesize{\text{is any real number\}}}&\{y\in\mathbb{R}|~y\geq{-4}\}\\\\\hline

\end{array}Interval NotationSet Notation​Domain(−∞,∞){x∈R∣ x is any real number}​Range[−4,∞){y∈R∣ y≥−4}​​

y=xy=\sqrt{x}y=x​

1. Sketch the function:
  


2. Identify any maximums or minimums that could restrict the input & output values

xyMinimum00MaximumNoneNone\begin{array}
{|c|c|c|}
\hline
&x&y\\\hline\\
\text{Minimum}&0&0\\\\\hline
\\\text{Maximum}&\text{None}&\text{None}\\\\\hline
\end{array}MinimumMaximum​x0None​y0None​​


3. Express the domain & range in interval & set notation

DomainRangeInterval Notation[0,∞)[0,∞)Set Notation{x∈R∣ x≥0}{y∈R∣ y≥0}\begin{array}
{|l|c|c|}\hline
\text{}&\text{Domain}&\text{Range}\\\hline\\

\text{Interval Notation}&[0,\infin)&[0,\infin)\\\\\hline\\

\text{Set Notation}&\{x\in\mathbb{R}|~x\geq{0}\}&\{y\in\mathbb{R}|~y\geq{0}\}\\\\\hline

\end{array}Interval NotationSet Notation​Domain[0,∞){x∈R∣ x≥0}​Range[0,∞){y∈R∣ y≥0}​​

Find the range of y=−2x2+5y=-2x^2+5y=−2x2+5. 

A) (−∞,5](-\infin,5](−∞,5]

B)[5,∞)[5,\infin)[5,∞)

C) {y∈R∣ y≤5}\{y\in\mathbb{R}|~y\leq{5}\}{y∈R∣ y≤5}

I don't know

Match the first two functions with their domain, and the second two function with their range.

A.

[0,∞)[0, \infin)[0,∞)

B.

[1,∞)[1,\infin)[1,∞)

C.

{y∈R∣ y is all real numbers}\{y\in\mathbb{R}|~y~{\footnotesize\text{is all real numbers}}\}{y∈R∣ y is all real numbers}

D.

(−∞,−4](-\infin,-4](−∞,−4]

y=x+1y=\sqrt{x}+1y=x​+1

y=x−1y=\sqrt{x-1}y=x−1​

y=5x+1y=5x+1y=5x+1

y=−x2−4y=-x^2-4y=−x2−4

I don't know

Extra Practice

Domain and Range

Determine the domain for the function y=4xx2+5x+6y=\dfrac{4x}{x^2+5x+6}y=x2+5x+64x​.

Vertical & Horizontal Translations

Practice: Vertical & Horizontal Translations
Let f(x)=x2f(x)=x^2f(x)=x2. 
True or False:

Vertical & Horizontal Translations

Practice: Vertical & Horizontal Translations
Let f(x)=x3f(x)=x^3f(x)=x3. 
True or False:

Domain and Range

Determine the domain for the function y=4xx2+5x+6y=\dfrac{4x}{x^2+5x+6}y=x2+5x+64x​.

Practice: Domain & Range
Find the domain & range of the following function shown below:

Practice: Domain & Range
Find the domain & range of the following function shown below:

Domain & Range

Practice: Domain & Range
Match the function with its appropriate domain or range.

Domain & Range

Practice: Domain & Range
Match the function with its appropriate domain or range.

Domain & Range

Practice: Domain & Range
Find the range of y=−(x+5)2+1y=-(x+5)^2+1y=−(x+5)2+1. 

Domain & Range

Practice: Domain & Range
Find the range of y=6x2−10y=6x^2-10y=6x2−10. 