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Let's calculate the inverse functions for the given functions:

### a) \( f(x) = \sqrt{10 - 2x} \)
To find the inverse:
1. Replace \( f(x) \) with \( y \):  
   \( y = \sqrt{10 - 2x} \)
2. Square both sides to eliminate the square root:  
   \( y^2 = 10 - 2x \)
3. Solve for \( x \):  
   \( 2x = 10 - y^2 \)  
   \( x = \frac{10 - y^2}{2} \)
4. Swap \( x \) and \( y \) to find the inverse:  
   \( f^{-1}(x) = \frac{10 - x^2}{2} \)

### b) \( g(x) = e^{x^3} \)
To find the inverse:
1. Replace \( g(x) \) with \( y \):  
   \( y = e^{x^3} \)
2. Apply the natural logarithm (ln) to both sides:  
   \( \ln y = x^3 \)
3. Solve for \( x \):  
   \( x = \sqrt[3]{\ln y} \)
4. Swap \( x \) and \( y \) to get the inverse:  
   \( g^{-1}(x) = \sqrt[3]{\ln x} \)

### c) \( h(x) = \ln x + 3 \)
To find the inverse:
1. Replace \( h(x) \) with \( y \):  
   \( y = \ln x + 3 \)
2. Subtract 3 from both sides:  
   \( y - 3 = \ln x \)
3. Exponentiate both sides (apply \( e \) to both sides):  
   \( x = e^{y - 3} \)
4. Swap \( x \) and \( y \):  
   \( h^{-1}(x) = e^{x - 3} \)

### d) \( s(x) = 2x^3 + 3 \)
To find the inverse:
1. Replace \( s(x) \) with \( y \):  
   \( y = 2x^3 + 3 \)
2. Subtract 3 from both sides:  
   \( y - 3 = 2x^3 \)
3. Divide by 2:  
   \( \frac{y - 3}{2} = x^3 \)
4. Take the cube root of both sides:  
   \( x = \sqrt[3]{\frac{y - 3}{2}} \)
5. Swap \( x \) and \( y \):  
   \( s^{-1}(x) = \sqrt[3]{\frac{x - 3}{2}} \)

### e) \( t(x) = \frac{4x - 1}{2x + 3} \)
To find the inverse:
1. Replace \( t(x) \) with \( y \):  
   \( y = \frac{4x - 1}{2x + 3} \)
2. Multiply both sides by \( 2x + 3 \) to get rid of the denominator:  
   \( y(2x + 3) = 4x - 1 \)
3. Expand the left-hand side:  
   \( 2xy + 3y = 4x - 1 \)
4. Group all \( x \)-terms on one side:  
   \( 2xy - 4x = -1 - 3y \)
   \( x(2y - 4) = -1 - 3y \)
5. Solve for \( x \):  
   \( x = \frac{-1 - 3y}{2y - 4} \)
6. Swap \( x \) and \( y \):  
   \( t^{-1}(x) = \frac{-1 - 3x}{2x - 4} \)

These are the inverse functions of the given ones.

Would you like further clarification on any step or more details? Here are some related questions for practice:

1. What is the domain and range of the inverse of \( f(x) = \sqrt{10 - 2x} \)?
2. How do you determine if a function has an inverse?
3. Can every exponential function have an inverse? Why or why not?
4. What conditions must be met for a rational function like \( t(x) \) to have an inverse?
5. How can we check if the inverse function we calculated is correct?

**Tip:** The derivative of a function can help determine if it is one-to-one, which is necessary for a function to have an inverse.

Calcule a função inversa para as funções a) f(x) = sqrt(10 - 2x), b) g(x) = e^(x^3), c) h(x) = ln(x) + 3, d) s(x) = 2x^3 + 3, e) t(x) = (4x - 1)/(2x + 3)

Step-by-step solution for calculating inverse functions for f(x) = sqrt(10 - 2x), g(x) = e^(x^3), h(x) = ln(x) + 3, s(x) = 2x^3 + 3, and t(x) = (4x - 1)/(2x + 3). This problem covers inverse function rules, logarithmic and exponential relationships, and rational function inverses.

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