Question

which of the following functions has domain R the set of real numbers (√x,|x|,1/x,tan x​

Answer 1

The function $|x|$ has a domain $R$ as the set of real numbers.

Step-by-step explanation:

• Positive real numbers, including zero, make up the domain of the function $\sqrt{x}$.
• Since the square root function only returns a real value for positive real numbers as input.
• If the function's input is a negative real number, it will output an imaginary number.
• For any real numbers, the modulus function is defined.
• But only positive real numbers, including zero, are inside its range.
• For any real values, other than those where $q$ is zero, a function of the form  $\frac{p}{q}$ is defined.
• The value  $\frac{p}{q}$  is not defined when $q$ is zero.
• Therefore, any real integers other than zero are the domain of the function $\frac{1}{x}$.
• $tan x$ is equal to the product of $sin x$ and $cos x$, or $tanx= \frac{sinx}{cosx}$.
• With the exception of the locations where $cosx$ is zero, its value is defined for all real values.
• Since $n$ is an integer, the domain of $tanx$ is $R-[(2n+1)\frac{\pi }{2} ]$.

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Answer 2

Answer:

The function $|x|$ has the domain $R$, the set of real numbers.

Step-by-step explanation:

Domain: The set of numbers for which the given function is defined is called domain.

Real numbers: The collection of all rational and irrational numbers is called real numbers.

1. $\sqrt{x}$

For $x\geq 0$, the function $\sqrt{x}$ is a real number.

For $x < 0$, the function $\sqrt{x}$ is a complex number.

Thus, $\sqrt{x}$ does not has the domain $R$.

2. $|x|$

For all real values of $x$, the modulus function $|x|$ is defined.

Thus, $|x|$ has the domain $R$.

3. $\frac{1}{x}$

Observe that the function $1/x$ at $x=0$ is not defined.

Thus, $\frac{1}{x}$ has the domain $R-${$0$} not the domain $R$.

4. $tanx$

We know that $tanx=\frac{sinx}{cosx}$.

Since the trigonometric function $cosx$ is zero for $(2n+1)\frac{\pi}{2}$, $n$∈$Z$.

⇒ the function $tanx$ is not defined at $x=(2n+1)\frac{\pi}{2}$

Thus, $tanx$ does not has the domain $R$.

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