What is the Range of the Function Graphed Below

For the
constant function
[latex]f\left(ten\right)=c[/latex], the domain consists of all existent numbers; in that location are no restrictions on the input. The but output value is the abiding [latex]c[/latex], and so the range is the prepare [latex]\left\{c\right\}[/latex] that contains this single element. In interval notation, this is written as [latex]\left[c,c\right][/latex], the interval that both begins and ends with [latex]c[/latex].

For the
identity part
[latex]f\left(ten\right)=10[/latex], in that location is no brake on [latex]ten[/latex]. Both the domain and range are the set of all real numbers.

For the
absolute value function
[latex]f\left(10\correct)=|10|[/latex], there is no brake on [latex]x[/latex]. Notwithstanding, because absolute value is divers as a distance from 0, the output can only be greater than or equal to 0.

For the
quadratic function
[latex]f\left(x\correct)={10}^{2}[/latex], the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.

For the
cubic function
[latex]f\left(x\right)={ten}^{3}[/latex], the domain is all real numbers considering the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, and then the domain and range include all real numbers.

For the
reciprocal office
[latex]f\left(ten\right)=\frac{1}{x}[/latex], we cannot divide by 0, so nosotros must exclude 0 from the domain. Further, i divided by any value can never be 0, so the range also volition not include 0. In set-builder notation, we could likewise write [latex]\left\{x|\text{ }x\ne 0\right\}[/latex], the set of all real numbers that are not nothing.

For the
reciprocal squared role
[latex]f\left(10\correct)=\frac{1}{{ten}^{2}}[/latex], we cannot divide by [latex]0[/latex], then we must exclude [latex]0[/latex] from the domain. There is also no [latex]x[/latex] that can requite an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, then the range includes only positive numbers.

For the
foursquare root part
[latex]f\left(x\correct)=\sqrt[]{x}[/latex], we cannot take the foursquare root of a negative real number, then the domain must be 0 or greater. The range also excludes negative numbers because the square root of a positive number [latex]x[/latex] is divers to be positive, even though the square of the negative number [latex]-\sqrt{10}[/latex] also gives us [latex]10[/latex].

For the
cube root function
[latex]f\left(x\correct)=\sqrt[3]{ten}[/latex], the domain and range include all real numbers. Annotation that there is no trouble taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (information technology is an odd office).