Inverses

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What Are Inverses?

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The inverse of a function undoes the action of that function.
For example, since \( x^2 \) reverts \( \sqrt{x} \) back to \( x \), and vice versa, they are inverse functions.
The inverse of \( f(x) \) is denoted \( f^{-1}(x) \).

Exercise 1: What is the inverse of \( x + 1 \)?
Exercise 2: What is the inverse of \( 5x \)?
Exercise 3: What is the inverse of \( x^3 \)?

Finding Inverses

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To find the inverse of a function:

1. Replace \( f(x) \) with \( y \).
2. Swap \( x \) and \( y \).
3. Solve for the new \( y \).

For example:

\[ \begin{aligned} y &= x^2 + 1 \\ x &= y^2 + 1 \\ x - 1 &= y^2 \\ y &= \sqrt{x - 1} \end{aligned} \]

So the inverse of \( f(x) = x^2 + 1 \) is \( f^{-1}(x) = \sqrt{x - 1} \).

Exercise 4: What is the inverse of \( 4x + 2 \)?
Exercise 5: What is the inverse of \( 5x^2 + 6 \)?
Exercise 6: What is the inverse of \( \frac{1}{x} \)?

Verifying Inverses

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Since an inverse undoes its original function, you can verify an inverse by using composition.
If \( f^{-1}(x) \) is truly the inverse of \( f(x) \), then:

• \( f(f^{-1}(x)) = x \)
• \( f^{-1}(f(x)) = x \)

Exercise 7: In our previous example, does \( f(f^{-1}(x)) = x \)?
Exercise 8: In our previous example, does \( f^{-1}(f(x)) = x \)?
Exercise 9: Do the two previous questions always verify an inverse?

Review

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Exercise 10: If \( f(1) = 2 \), what is the value of \( f^{-1}(2) \)?
Exercise 11: If \( f(x) = \frac{\prod_{n=1}^5 \frac{n}{n + 1}}{\sum_{n=1}^{x} \frac{e}{n}} \), what is the value of \( f^{-1}(f(546)) \)?
Exercise 12: What is the inverse of \( \frac{2}{7x^2} \)?

Credits

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All explanations and problems written by Adrian Hernandez Vega, unless otherwise noted.

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