Sure, let's identify the functions that have the given properties:
1. The domain is all real numbers for [tex]\(\square\)[/tex]:
For this property, functions that have all real numbers as their domain include any polynomial, sine, cosine, and exponential functions, among others. These functions are defined and continuous for every real number [tex]\(x\)[/tex].
2. An [tex]\(x\)[/tex]-intercept is [tex]\((\pi, 0)\)[/tex] for [tex]\(\square\)[/tex]:
An [tex]\(x\)[/tex]-intercept [tex]\((\pi, 0)\)[/tex] means that the function passes through the point where [tex]\(x = \pi\)[/tex] and [tex]\(y = 0\)[/tex]. One such function that satisfies this condition is [tex]\( \sin(x) \)[/tex]. The sine function reaches zero at [tex]\(x = \pi\)[/tex], hence having an [tex]\(x\)[/tex]-intercept at [tex]\((\pi, 0)\)[/tex].
3. The minimum value is -1 for [tex]\(\square\)[/tex]:
Functions that have a minimum value of -1 include [tex]\( \sin(x) \)[/tex] and [tex]\( \cos(x) \)[/tex]. Both sine and cosine functions oscillate between -1 and 1, achieving the minimum value of -1 at specific points.
4. An [tex]\(x\)[/tex]-intercept is [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex] for [tex]\(\square\)[/tex]:
An [tex]\(x\)[/tex]-intercept at [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex] means that the function passes through the point where [tex]\(x = \frac{\pi}{2}\)[/tex] and [tex]\(y = 0\)[/tex]. The function [tex]\( \tan(x) \)[/tex] has this property because tangent reaches zero at [tex]\(x = \frac{\pi}{2}\)[/tex], hence producing an [tex]\(x\)[/tex]-intercept at [tex]\((\frac{\pi}{2}, 0)\)[/tex].
So, summarizing based on the properties:
- The domain is all real numbers for any polynomial, sine, cosine, exponential functions, etc.
- An [tex]\(x\)[/tex]-intercept is [tex]\((\pi, 0)\)[/tex] for [tex]\(\sin(x)\)[/tex]
- The minimum value is -1 for [tex]\(\sin(x), \cos(x)\)[/tex]
- An [tex]\(x\)[/tex]-intercept is [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex] for [tex]\(\tan(x)\)[/tex]
