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Let's work through the inverses of the given functions.

### (a) \( f(x) = x^2 + 3x - 1 \), \( x > -\frac{3}{2} \)
To find the inverse:
1. Set \( y = x^2 + 3x - 1 \).
2. Swap \( x \) and \( y \) and solve for \( y \):
   \[
   x = y^2 + 3y - 1
   \]
   This quadratic equation in \( y \) is tricky to solve directly, but it's a standard method. We can use the quadratic formula:
   \[
   y = \frac{-3 \pm \sqrt{3^2 - 4(1)(-1+x)}}{2}
   \]
   Solving this would yield the inverse, which is quite complex.

### (b) \( f(x) = -4x + 8 \)
1. Set \( y = -4x + 8 \).
2. Swap \( x \) and \( y \):
   \[
   x = -4y + 8
   \]
3. Solve for \( y \):
   \[
   y = \frac{8 - x}{4}
   \]
Thus, the inverse function is:
\[
f^{-1}(x) = \frac{8 - x}{4}
\]

### (c) \( f(x) = x^3 - 2 \)
1. Set \( y = x^3 - 2 \).
2. Swap \( x \) and \( y \):
   \[
   x = y^3 - 2
   \]
3. Solve for \( y \):
   \[
   y = \sqrt[3]{x + 2}
   \]
Thus, the inverse function is:
\[
f^{-1}(x) = \sqrt[3]{x + 2}
\]

### (d) \( f(x) = e^x \)
1. Set \( y = e^x \).
2. Swap \( x \) and \( y \):
   \[
   x = e^y
   \]
3. Solve for \( y \) by taking the natural logarithm on both sides:
   \[
   y = \ln(x)
   \]
Thus, the inverse function is:
\[
f^{-1}(x) = \ln(x)
\]

### (e) \( f(x) = \sqrt{x + 1} \)
1. Set \( y = \sqrt{x + 1} \).
2. Swap \( x \) and \( y \):
   \[
   x = \sqrt{y + 1}
   \]
3. Square both sides to solve for \( y \):
   \[
   x^2 = y + 1 \quad \Rightarrow \quad y = x^2 - 1
   \]
Thus, the inverse function is:
\[
f^{-1}(x) = x^2 - 1
\]

### (f) \( f(x) = \ln(x + 2) \)
1. Set \( y = \ln(x + 2) \).
2. Swap \( x \) and \( y \):
   \[
   x = \ln(y + 2)
   \]
3. Exponentiate both sides to get rid of the logarithm:
   \[
   e^x = y + 2 \quad \Rightarrow \quad y = e^x - 2
   \]
Thus, the inverse function is:
\[
f^{-1}(x) = e^x - 2
\]

Would you like more details on any of these steps?

### Relative Questions:
1. How can we determine if a function has an inverse?
2. What are the necessary conditions for a function to have an inverse?
3. How does the domain of a function affect its inverse?
4. Can all quadratic functions have inverses?
5. How do you verify if two functions are inverses of each other?

### Tip:
Always check the domain and range of both the function and its inverse, as they often provide important constraints, especially for square roots or logarithms.

Find the inverse function of f(x) for the given functions: (a) f(x) = x^2 + 3x - 1, x > -3/2; (b) f(x) = -4x + 8; (c) f(x) = x^3 - 2; (d) f(x) = e^x; (e) f(x) = sqrt(x + 1); (f) f(x) = ln(x + 2).

Step-by-step solutions to find the inverse functions of various mathematical functions, including quadratic, cubic, exponential, and logarithmic functions. Learn how to solve using formulas like the quadratic equation, logarithmic properties, and cubic roots.

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