To determine the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-intercepts for the given function [tex]\( p(x) = -|x| + 1 \)[/tex], we need to follow these steps:
### Part 1: [tex]\( x \)[/tex]-intercepts
The [tex]\( x \)[/tex]-intercepts occur where the function crosses the [tex]\( x \)[/tex]-axis, i.e., where [tex]\( p(x) = 0 \)[/tex].
1. Set the function equal to zero:
[tex]\[
-|x| + 1 = 0
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
-|x| = -1
\][/tex]
3. Multiply both sides by -1:
[tex]\[
|x| = 1
\][/tex]
4. Since the absolute value [tex]\( |x| \)[/tex] can be either positive or negative, we get two [tex]\( x \)[/tex]-intercepts:
[tex]\[
x = 1 \quad \text{and} \quad x = -1
\][/tex]
So, the [tex]\( x \)[/tex]-intercepts are:
[tex]\[
(1, 0) \quad \text{and} \quad (-1, 0)
\][/tex]
### Part 2: [tex]\( y \)[/tex]-intercept
The [tex]\( y \)[/tex]-intercept occurs where the function crosses the [tex]\( y \)[/tex]-axis, i.e., where [tex]\( x = 0 \)[/tex]:
1. Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[
p(0) = -|0| + 1
\][/tex]
2. Simplify:
[tex]\[
p(0) = 0 + 1 = 1
\][/tex]
So, the [tex]\( y \)[/tex]-intercept is:
[tex]\[
(0, 1)
\][/tex]
### Final Answers:
1. [tex]\( x \)[/tex]-intercepts:
[tex]\[
(1, 0), (-1, 0)
\][/tex]
2. [tex]\( y \)[/tex]-intercept:
[tex]\[
(0, 1)
\][/tex]
