Let's work through the given properties step-by-step and identify the functions that satisfy each of them:
1. The domain is all real numbers.
- A function with a domain of all real numbers can be a variety of common functions including linear, quadratic, polynomial, exponential, etc.
- One simple example is the quadratic function:
[tex]\[
f(x) = x^2
\][/tex]
This function is defined for all real numbers.
2. An [tex]\(x\)[/tex]-intercept is [tex]\((\pi, 0)\)[/tex].
- An [tex]\(x\)[/tex]-intercept is a point where the function crosses the x-axis, meaning the output of the function is zero at that point:
[tex]\[
f(\pi) = 0
\][/tex]
- A function that satisfies this condition is the sine function, since:
[tex]\[
\sin(\pi) = 0
\][/tex]
Therefore, the function is:
[tex]\[
f(x) = \sin(x)
\][/tex]
3. The minimum value is -1.
- A function whose minimum value is -1 achieves this value somewhere in its domain.
- The sine function has a minimum value of -1, as the sine of an angle ranges from -1 to 1:
[tex]\[
\min(\sin(x)) = -1
\][/tex]
Thus, the function is:
[tex]\[
f(x) = \sin(x)
\][/tex]
4. An [tex]\(x\)[/tex]-intercept is [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex].
- Similarly, to have an [tex]\(x\)[/tex]-intercept at [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex], the function must satisfy:
[tex]\[
f\left(\frac{\pi}{2}\right) = 0
\][/tex]
- The cosine function meets this condition because:
[tex]\[
\cos\left(\frac{\pi}{2}\right) = 0
\][/tex]
Therefore, the function is:
[tex]\[
f(x) = \cos(x)
\][/tex]
Summarizing all the functions that satisfy the given properties:
1. For all real numbers domain: [tex]\(x^2\)[/tex]
2. For [tex]\(x\)[/tex]-intercept at [tex]\((\pi, 0)\)[/tex]: [tex]\(\sin(x)\)[/tex]
3. For the minimum value of -1: [tex]\(\sin(x)\)[/tex]
4. For [tex]\(x\)[/tex]-intercept at [tex]\(\left(\frac{\pi}{2}, 0\right)\)[/tex]: [tex]\(\cos(x)\)[/tex]
Thus, the functions that meet each specific property are:
[tex]\[
(\text{x}^2, \; \sin(x), \; \sin(x), \; \cos(x))
\][/tex]
