Sure, let's determine the domains of the given functions step by step.
1. Function [tex]\( p(t) = -9t - 5 \)[/tex]:
- This is a linear function.
- Linear functions are defined for all real numbers since there are no restrictions or potential undefined points.
- Therefore, the domain of [tex]\( p(t) \)[/tex] is all real numbers.
- In interval notation, this is written as [tex]\( \mathbf{(-\infty, \infty)} \)[/tex].
2. Function [tex]\( f(x) = x^2 + 3 \)[/tex]:
- This is a quadratic function.
- Quadratic functions are also defined for all real numbers as they have no restrictions on the input values.
- Therefore, the domain of [tex]\( f(x) \)[/tex] is all real numbers.
- In interval notation, this is written as [tex]\( \mathbf{(-\infty, \infty)} \)[/tex].
3. Function [tex]\( h(x) = \sqrt{x - 4} \)[/tex]:
- This is a square root function.
- Square root functions are defined only when the expression inside the square root is non-negative.
- Therefore, the expression [tex]\( x - 4 \)[/tex] must be greater than or equal to zero.
- Solving [tex]\( x - 4 \geq 0 \)[/tex], we get [tex]\( x \geq 4 \)[/tex].
- Therefore, the domain of [tex]\( h(x) \)[/tex] is all real numbers greater than or equal to 4.
- In interval notation, this is written as [tex]\( \mathbf{[4, \infty)} \)[/tex].
4. Function [tex]\( g(x) = \frac{1}{x + 7} \)[/tex]:
- This is a rational function.
- Rational functions are undefined where the denominator is zero.
- Therefore, the expression [tex]\( x + 7 \)[/tex] must not be zero.
- Solving [tex]\( x + 7 \neq 0 \)[/tex], we get [tex]\( x \neq -7 \)[/tex].
- Therefore, the domain of [tex]\( g(x) \)[/tex] is all real numbers except [tex]\(-7\)[/tex].
- In interval notation, this is written as [tex]\( \mathbf{(-\infty, -7) \cup (-7, \infty)} \)[/tex].
To summarize, the domains for the given functions in interval notation are:
- [tex]\( p(t) = -9t - 5 \)[/tex] : [tex]\(\mathbf{(-\infty, \infty)}\)[/tex]
- [tex]\( f(x) = x^2 + 3 \)[/tex] : [tex]\(\mathbf{(-\infty, \infty)}\)[/tex]
- [tex]\( h(x) = \sqrt{x - 4} \)[/tex] : [tex]\(\mathbf{[4, \infty)}\)[/tex]
- [tex]\( g(x) = \frac{1}{x + 7} \)[/tex] : [tex]\(\mathbf{(-\infty, -7) \cup (-7, \infty)}\)[/tex]
These are the domains for each of the given functions.
