Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Discover precise answers to your questions from a wide range of experts on our user-friendly Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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.
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.
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
