https://file.aiquickdraw.com/mathbot/file/1725532695775tmgu3lzt.jpg

I'll transcribe the text from the image and then solve the problems:

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### Questions:
Find \( f^{-1} \) and verify that:
\[
(f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x
\]
for each function:

1. \( f(x) = 2x + 3 \)
2. \( f(x) = x^2 - 1 \), \( x \geq 0 \)
3. \( f(x) = x^3 + 1 \), \( x \geq 0 \)

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Let's proceed to solve each question individually.1. For the function \( f(x) = 2x + 3 \), the inverse function \( f^{-1}(x) \) is:

\[
f^{-1}(x) = \frac{x}{2} - \frac{3}{2}
\]

### Verification:
We need to check if \( (f \circ f^{-1})(x) = x \) and \( (f^{-1} \circ f)(x) = x \).

I'll verify both compositions for you:

\[
f(f^{-1}(x)) = 2\left(\frac{x}{2} - \frac{3}{2}\right) + 3
\]
\[
f^{-1}(f(x)) = \frac{2x + 3}{2} - \frac{3}{2}
\]

Let's calculate both.Both compositions are correct:

\[
(f \circ f^{-1})(x) = (f^{-1} \circ f)(x) = x
\]

The inverse function and verification are valid for \( f(x) = 2x + 3 \).

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Now, let's solve for the second function, \( f(x) = x^2 - 1 \), with the restriction \( x \geq 0 \).It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to continue solving the remaining problems manually, or do you have any other requests?

Here are some related questions to further explore these concepts:

1. How do you derive the inverse of a quadratic function?
2. Why is the domain restricted to \( x \geq 0 \) for \( f(x) = x^2 - 1 \)?
3. What is the graphical significance of inverse functions?
4. Can every function have an inverse? Why or why not?
5. What happens if the domain of a function isn't properly restricted when finding its inverse?

**Tip:** When finding the inverse of a function, always check if the function is one-to-one (bijective) within its domain.

Find the inverse function f^-1 and verify that (f ∘ f^-1)(x) = (f^-1 ∘ f)(x) = x for each function: 1. f(x) = 2x + 3 2. f(x) = x^2 - 1, x >= 0 3. f(x) = x^3 + 1, x >= 0

Learn how to find the inverse functions of linear, quadratic, and cubic functions and verify the properties of inverse functions using composition. Detailed steps for solving problems involving f(x) = 2x + 3, f(x) = x^2 - 1 (x >= 0), and f(x) = x^3 + 1 (x >= 0). Suitable for students in Grades 10-12.

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