Questions: What of the following functions have domain x ∈ R (i.e no restrictions on their domain)? Select ALL that apply. 1/(sqrt(x-10)) e^(x-10)+3 x^6+10 sqrt(x+10)

Solution Steps

To determine which functions have a domain of all real numbers, we need to identify any restrictions on the domain for each function. Specifically, we look for divisions by zero or square roots of negative numbers, as these create restrictions.

1. For the function \(\frac{1}{\sqrt{x-10}}\), the domain is restricted because the expression under the square root must be positive, and the denominator cannot be zero.
2. The function \(e^{x-10}+3\) has no restrictions, as the exponential function is defined for all real numbers.
3. The function \(x^{6}+10\) is a polynomial, which is defined for all real numbers.
4. The function \(\sqrt{x+10}\) is restricted because the expression under the square root must be non-negative.

To determine which functions have a domain of all real numbers, we need to analyze each function for any restrictions:

1. Function: \(\frac{1}{\sqrt{x-10}}\)

   • Restriction: The expression under the square root, \(x-10\), must be positive, and the denominator cannot be zero. Therefore, \(x > 10\).
   • Domain: \(x > 10\)

2. Function: \(e^{x-10} + 3\)

   • Restriction: The exponential function is defined for all real numbers.
   • Domain: \(x \in \mathbb{R}\)

3. Function: \(x^6 + 10\)

   • Restriction: This is a polynomial function, which is defined for all real numbers.
   • Domain: \(x \in \mathbb{R}\)

4. Function: \(\sqrt{x+10}\)

   • Restriction: The expression under the square root, \(x+10\), must be non-negative. Therefore, \(x \geq -10\).
   • Domain: \(x \geq -10\)

From the analysis above, we identify the functions that have a domain of all real numbers:

• \(e^{x-10} + 3\) has a domain of \(x \in \mathbb{R}\).
• \(x^6 + 10\) has a domain of \(x \in \mathbb{R}\).

Final Answer