show that the function is one to one and then graph its inverse. Show all the written work for this problem to be worked out and only put that.1.) y= cos x , 0≤x≤pi

To show that the function \( y = \cos(x) \) for \( 0 \leq x \leq \pi \) is one-to-one and then graph its inverse, we follow these steps:

### Step 1: Show that \( y = \cos(x) \) is One-to-One on \( [0, \pi] \)

A function is one-to-one (injective) if every element of its range is mapped from a unique element of its domain. Mathematically, a function \( f(x) \) is one-to-one if, for any \( x_1, x_2 \in [0, \pi] \), \( f(x_1) = f(x_2) \) implies that \( x_1 = x_2 \).

For \( y = \cos(x) \), we need to show that:

\[
\cos(x_1) = \cos(x_2) \implies x_1 = x_2
\]

The cosine function is strictly decreasing on the interval \( [0, \pi] \). This means that if \( x_1 < x_2 \), then \( \cos(x_1) > \cos(x_2) \). Therefore, it is impossible for \( \cos(x_1) = \cos(x_2) \) unless \( x_1 = x_2 \). Thus, \( \cos(x) \) is one-to-one on the interval \( [0, \pi] \).

### Step 2: Find the Inverse of \( y = \cos(x) \)

To find the inverse, we need to solve for \( x \) in terms of \( y \):

\[
y = \cos(x) \implies x = \arccos(y)
\]

Thus, the inverse function is:

\[
x = \cos^{-1}(y)
\]

or

\[
y = \arccos(x)
\]

### Step 3: Graph the Inverse Function \( y = \arccos(x) \)

To graph the inverse function \( y = \arccos(x) \), we note the following:

- The range of \( \cos(x) \) on \( [0, \pi] \) is \( [-1, 1] \), which becomes the domain of the inverse function.
- The domain of \( \cos(x) \) on \( [0, \pi] \) is \( [0, \pi] \), which becomes the range of the inverse function.

The graph of \( y = \arccos(x) \) will be a reflection of the graph of \( y = \cos(x) \) over the line \( y = x \).

Here is the graph of \( y = \arccos(x) \):

\[
\text{Graph of } y = \arccos(x) \text{ on } [-1, 1].
\]

1. The function passes through points \( (1, 0) \), \( (0, \frac{\pi}{2}) \), and \( (-1, \pi) \).
2. It is decreasing and concave on its domain.

The graph looks like an arc from \( (1, 0) \) to \( (-1, \pi) \), decreasing from left to right.

These steps show that the function is one-to-one and how its inverse is graphed.

Would you like a visual representation of the graph? 

### 5 Questions to Explore Further:
1. What are the properties of the cosine function on the interval \( [0, \pi] \)?
2. How do you derive the range of the inverse cosine function?
3. Can you explain the difference between the principal value of \( \arccos(x) \) and \( \arcsin(x) \)?
4. How would the graph change if we considered \( y = \cos(x) \) on a different interval?
5. What are some real-life applications of inverse trigonometric functions?

**Tip:** When finding inverses of trigonometric functions, always pay attention to the interval over which the original function is defined to ensure that it is one-to-one.

show that the function is one to one and then graph its inverse. Show all the written work for this problem to be worked out and only put that. 1.) y= cos x , 0≤x≤pi

Learn how to prove that the function y = cos(x) is one-to-one on the interval [0, π] and how to graph its inverse function y = arccos(x). This problem covers key trigonometric concepts and is suitable for students in Grades 11-12.

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