Let’s find the inverses for the given functions step by step.

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1. For $f(x) = 5x - 2$:

To find the inverse:

1. Replace $f(x)$ with $y$:
   $y = 5x - 2$.
2. Swap $x$ and $y$:
   $x = 5y - 2$.
3. Solve for $y$:
   $x + 2 = 5y \implies y = \frac{x + 2}{5}.$

Inverse function:
$f^{-1}(x) = \frac{x + 2}{5}.$

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2. For $f(x) = \frac{1}{2}x + 4$:

To find the inverse:

1. Replace $f(x)$ with $y$:
   $y = \frac{1}{2}x + 4$.
2. Swap $x$ and $y$:
   $x = \frac{1}{2}y + 4$.
3. Solve for $y$:
   $x - 4 = \frac{1}{2}y \implies y = 2(x - 4).$

Inverse function:
$f^{-1}(x) = 2(x - 4).$

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3. For $f(x) = 2(x - 3)^2 + 1$:

To find the inverse:

1. Replace $f(x)$ with $y$:
   $y = 2(x - 3)^2 + 1$.
2. Swap $x$ and $y$:
   $x = 2(y - 3)^2 + 1$.
3. Solve for $y$:
   $x - 1 = 2(y - 3)^2 \implies \frac{x - 1}{2} = (y - 3)^2.$ Taking the square root (± for both branches): $y - 3 = \pm \sqrt{\frac{x - 1}{2}} \implies y = 3 \pm \sqrt{\frac{x - 1}{2}}.$

Since the original function is not one-to-one, its inverse has two branches:
$f^{-1}(x) = 3 + \sqrt{\frac{x - 1}{2}} \quad \text{and} \quad f^{-1}(x) = 3 - \sqrt{\frac{x - 1}{2}}.$

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Summary of Inverses:

1. $f^{-1}(x) = \frac{x + 2}{5}$
2. $f^{-1}(x) = 2(x - 4)$
3. $f^{-1}(x) = 3 \pm \sqrt{\frac{x - 1}{2}}$

Would you like further clarification or detailed steps for any of these?

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Related Questions:

1. How can we determine whether a function has an inverse before solving?
2. What is the graphical interpretation of a function and its inverse?
3. What happens if a function is not one-to-one? Can it still have an inverse?
4. How can we verify that our calculated inverse is correct?
5. Can a quadratic function ever have a single-valued inverse?

Tip: Always check if the original function is one-to-one by applying the horizontal line test before finding the inverse!