Select the correct answer. Which function has a domain of all real numbers? A.y=(x+2)^ 1/4 B.y=-x^ 1/2+5 C.y=-2(3x)^1/6 D.y=(2x)^1/3-7

The domain of a function is the set of input values for which the function is real and defined. In the other words, when you define the domain, you are indicating for which values x the function is real and defined.

Letter A - [tex]y=\left(x+2\right)^{^{\frac{1}{4}}\:}[/tex], as you have an even exponent, the results for y should be greater or equal than zero. Thus,

[tex]\left(x+2\right)^{^{\frac{1}{4}}\:}=\sqrt[4]{x+2}[/tex]

Therefore, the function domain is x [tex]\geq[/tex]-2.

Letter B - [tex]y=-x^{\frac{1}{2}}^{\:}+5[/tex], as you have an even exponent, the results for y should be greater or equal than zero. Thus,

[tex]-x^{\frac{1}{2}}^{\:}+5=-\sqrt{x} +5[/tex]

x [tex]\geq[/tex] 0

Therefore, the function domain is x [tex]\geq[/tex] 0.

Letter C - [tex]y=-2\left(3x\right)^{\frac{1}{6}}[/tex], as you have an even exponent, the results for y should be greater or equal than zero. Thus,

[tex]-2\left(3x\right)^{\frac{1}{6}}=-2*\sqrt[6]{3x}[/tex]

3x [tex]\geq[/tex] 0

x [tex]\geq[/tex] 0

Therefore, the function domain is x [tex]\geq[/tex] 0.

Letter D - [tex]y=\left(2x\right)^{\frac{1}{3}}-7[/tex], as you have an odd exponent, the results for y has no undefined points nor domain constraints. Thus,

[tex]\left(2x\right)^{\frac{1}{3}}-7=y=\sqrt[3]{2x} -7\\ \\[/tex]

Therefore, the function domain is [tex]-\infty \: < x < \infty \:[/tex].

Learn more about the domain here:

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