Sure, let's analyze the given properties step by step and identify the functions that satisfy each one.
1. The domain is all real numbers:
Functions that have the domain of all real numbers include:
- Polynomial functions: These functions are defined for all real numbers ([tex]\(-\infty, \infty\)[/tex]).
- Trigonometric functions: Functions like sine (`sin(x)`) and cosine (`cos(x)`) are also defined for all real numbers.
Therefore, the functions that satisfy this property are:
[tex]\[ \text{polynomial, trigonometric} \][/tex]
2. An [tex]\( x \)[/tex]-intercept is [tex]\( (\pi, 0) \)[/tex]:
- The [tex]\( x \)[/tex]-intercept refers to the points where the function crosses the x-axis, meaning the function value is zero at that point.
- The function that has an [tex]\( x \)[/tex]-intercept at [tex]\( \pi \)[/tex] is [tex]\( \sin(x) \)[/tex]. This is because [tex]\( \sin(\pi) = 0 \)[/tex].
Therefore, the function that satisfies this property is:
[tex]\[ \sin(x) \][/tex]
3. The minimum value is -1:
- For trigonometric functions, we look at [tex]\( \cos(x) \)[/tex], which achieves a minimum value of -1. The cosine function oscillates between -1 and 1, and reaches -1 at points [tex]\( x = \pi + 2k\pi \)[/tex] where [tex]\( k \)[/tex] is an integer.
Therefore, the function that satisfies this property is:
[tex]\[ \cos(x) \][/tex]
4. An [tex]\( x \)[/tex]-intercept is [tex]\( \left( \frac{\pi}{2}, 0 \right) \)[/tex]:
- The function that has an [tex]\( x \)[/tex]-intercept at [tex]\( \frac{\pi}{2} \)[/tex] is [tex]\( \cos(x) \)[/tex]. This is because [tex]\( \cos\left(\frac{\pi}{2}\right) = 0 \)[/tex].
Therefore, the function that satisfies this property is:
[tex]\[ \cos(x) \][/tex]
Summarizing all of the identified functions:
- The domain is all real numbers for: [tex]\[ \text{polynomial, trigonometric} \][/tex]
- An [tex]\( x \)[/tex]-intercept is [tex]\( (\pi, 0) \)[/tex] for: [tex]\[ \sin(x) \][/tex]
- The minimum value is -1 for: [tex]\[ \cos(x) \][/tex]
- An [tex]\( x \)[/tex]-intercept is [tex]\( \left( \frac{\pi}{2}, 0 \right) \)[/tex] for: [tex]\[ \cos(x) \][/tex]
