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Wize University Calculus 1 Textbook > Pre-Calculus (Review)

Domain & Range

Interval Notation vs. Set NotationPrevious Section
Operations with FunctionsNext Section
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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^2 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^2 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\}}
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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\geq5
  • Interval Notation: [5,∞)[5,\infty)
  • Set Notation: {x∈R∣ x≥5}\{x\in\mathbb{R}|~x\geq{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^2 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\geq0
  • Interval Notation: [0,∞)[0,\infin)
  • Set Notation: {y∈R∣ y≥0}\{y\in\mathbb{R}|~y\geq0\}
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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}
  • Interval Notation: [5,∞)[5,\infin)
  • Set Notation: {y∈R∣ y≥5}\{y\in\mathbb{R}|~y\geq{5}\}
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Example: Domain & Range

Find the domain & range of the following

  1. y=2x2−4y=2x^2-4
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}


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}

  1. y=xy=\sqrt{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}


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}
Find the range of y=−2x2+5y=-2x^2+5.
Match the first two functions with their domain, and the second two function with their range.
A.
(−∞,−4](-\infin,-4]
B.
{y∈R∣ y is all real numbers}\{y\in\mathbb{R}|~y~{\footnotesize\text{is all real numbers}}\}
C.
[1,∞)[1,\infin)
D.
[0,∞)[0, \infin)
y=x+1y=\sqrt{x}+1
y=x−1y=\sqrt{x-1}
y=5x+1y=5x+1
y=−x2−4y=-x^2-4
Extra Practice
Domain and Range
Determine the domain for the function y=4xx2+5x+6y=\dfrac{4x}{x^2+5x+6}.
Vertical & Horizontal Translations
Practice: Vertical & Horizontal Translations
Let f(x)=x2f(x)=x^2.
True or False:
Vertical & Horizontal Translations
Practice: Vertical & Horizontal Translations
Let f(x)=x3f(x)=x^3.
True or False:
Domain and Range
Determine the domain for the function y=4xx2+5x+6y=\dfrac{4x}{x^2+5x+6}.
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:
Limits at Infinity: Domain and Range
(Original question is written in long answer. Multiple choice has been used here for convenience)
Let f(x)=x2−6x−3\displaystyle f(x) = \frac{\sqrt{x^2 - 6}}{x - 3}.
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+1.
Domain & Range
Practice: Domain & Range
Find the range of y=6x2−10y=6x^2-10.
Domain & Range
Match the first two functions with their domain, and the second two function with their range.
Domain & Range
Find the range of y=−2x2+5y=-2x^2+5.
Domain and Range
Find the domain and range of f(x)=x+∣x∣xf\left(x\right)=\frac{x+\left|x\right|}{x}.