Step-by-step explanation:
Considering the given functions
- [tex]f\left(x\right)\:=\:2x^2\:-\:9x\:+\:2[/tex]
- [tex]g\left(x\right)\:=\:1\:-\:6x[/tex]
- [tex]h\left(x\right)\:=\:x^2\:-\:4[/tex]
As we know that
- the domain is the set of all possible input values of independent variable for which the function is defined.
- And range is the set of all possible output values after the substitution of the values in the domain interval.
- Also note that only real numbers could be included in the domain and range.
There are following two reasons why domains tend to be restricted.
- We can't divided any equation by 0 to prevent the function from becoming undefined.
- We can not take the square root of any negative real number to prevent the function from becoming undefined.
So, lets now check the functions one by one.
FOR THE FUNCTION [tex]f\left(x\right)\:=\:2x^2\:-\:9x\:+\:2[/tex]
Considering the function
- [tex]f\left(x\right)\:=\:2x^2\:-\:9x\:+\:2[/tex]
So,
[tex]\mathrm{Domain\:of\:}\:2x^2-9x+2\::\quad \begin{bmatrix}\mathrm{Solution:}\:&\:-\infty \:<x<\infty \\ \:\mathrm{Interval\:Notation:}&\:\left(-\infty \:,\:\infty \:\right)\end{bmatrix}[/tex]
As domain is the set of all real number. Therefore, there are no restrictions in the domain. It means the function is defined for all real numbers.
[tex]\mathrm{Range\:of\:}2x^2-9x+2:\quad \begin{bmatrix}\mathrm{Solution:}\:&\:f\left(x\right)\ge \:-\frac{65}{8}\:\\ \:\mathrm{Interval\:Notation:}&\:[-\frac{65}{8},\:\infty \:)\end{bmatrix}[/tex]
The graph is also shown. The green color Parabola is the graph of the function [tex]f\left(x\right)\:=\:2x^2\:-\:9x\:+\:2[/tex].
FOR THE FUNCTION [tex]g\left(x\right)\:=\:1\:-\:6x[/tex]
Considering the function
- [tex]g\left(x\right)\:=\:1\:-\:6x[/tex]
As
[tex]\mathrm{Domain\:of\:}\:1-6x\::\quad \begin{bmatrix}\mathrm{Solution:}\:&\:-\infty \:<x<\infty \\ \:\mathrm{Interval\:Notation:}&\:\left(-\infty \:,\:\infty \:\right)\end{bmatrix}[/tex]
As domain is the set of all real number. Therefore, there are no restrictions in the domain. It means the function is defined for all real numbers.
[tex]\mathrm{Range\:of\:}1-6x:\quad \begin{bmatrix}\mathrm{Solution:}\:&\:-\infty \:<f\left(x\right)<\infty \\ \:\mathrm{Interval\:Notation:}&\:\left(-\infty \:,\:\infty \:\right)\end{bmatrix}[/tex]
The graph is also shown. The blue color straight line is the graph of the function [tex]g\left(x\right)\:=\:1\:-\:6x[/tex].
FOR THE FUNCTION [tex]h\left(x\right)\:=\:x^2\:-\:4[/tex]
Considering the function
- [tex]h\left(x\right)\:=\:x^2\:-\:4[/tex]
As
[tex]\mathrm{Domain\:of\:}\:x^2-4\::\quad \begin{bmatrix}\mathrm{Solution:}\:&\:-\infty \:<x<\infty \\ \:\mathrm{Interval\:Notation:}&\:\left(-\infty \:,\:\infty \:\right)\end{bmatrix}[/tex]
As domain is the set of all real number. Therefore, there are no restrictions in the domain. It means the function is defined for all real numbers.
[tex]\mathrm{Range\:of\:}x^2-4:\quad \begin{bmatrix}\mathrm{Solution:}\:&\:f\left(x\right)\ge \:-4\:\\ \:\mathrm{Interval\:Notation:}&\:[-4,\:\infty \:)\end{bmatrix}[/tex]
The graph is also shown. The black color parabola is the graph of the function [tex]h\left(x\right)\:=\:x^2\:-\:4[/tex].
Keywords: equation, graph, function, domain, range, restricted values
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