Hey can you give me a step by step solution on how to calculate the inverse function in fractional form , and please include the formula on how to check that one.

To calculate the inverse of a function and express it in fractional form, follow these step-by-step instructions. For simplicity, I'll assume the function is given in terms of \( y = f(x) \). Here's a general guide:

### Step-by-Step Guide to Finding the Inverse Function

#### 1. **Write the Function in Terms of \( y \):**
   
   Suppose the original function is:

   \[
   y = f(x)
   \]

   Rewrite the function equation with \( y \) on the left side and the formula for \( f(x) \) on the right side. For example, let’s say:

   \[
   y = \frac{2x + 3}{5}
   \]

#### 2. **Switch \( x \) and \( y \):**

   To find the inverse, swap \( x \) and \( y \). This reflects the idea that for the inverse function \( f^{-1}(x) \), the roles of \( x \) and \( y \) are reversed:

   \[
   x = \frac{2y + 3}{5}
   \]

#### 3. **Solve for \( y \):**

   Now, solve for \( y \) in terms of \( x \). This involves isolating \( y \) on one side of the equation.

   - First, multiply both sides by 5 to eliminate the fraction:

     \[
     5x = 2y + 3
     \]

   - Subtract 3 from both sides:

     \[
     5x - 3 = 2y
     \]

   - Divide by 2 to isolate \( y \):

     \[
     y = \frac{5x - 3}{2}
     \]

   This is the inverse function:

   \[
   f^{-1}(x) = \frac{5x - 3}{2}
   \]

#### 4. **Express in Fractional Form:**

   If your result is not already in fractional form, ensure that the final expression is written as a fraction. In our example, the inverse is already a fraction:

   \[
   f^{-1}(x) = \frac{5x - 3}{2}
   \]

---

### Step-by-Step Guide to Verifying the Inverse

To verify that the function \( f(x) \) and its inverse \( f^{-1}(x) \) are correct, you must check both compositions: \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \). If both equal \( x \), then the inverse is correct.

#### 1. **Verify \( f(f^{-1}(x)) \):**

   Substitute \( f^{-1}(x) \) into \( f(x) \).

   - \( f(x) = \frac{2x + 3}{5} \), and \( f^{-1}(x) = \frac{5x - 3}{2} \).
   
   Now calculate:

   \[
   f\left( f^{-1}(x) \right) = f\left( \frac{5x - 3}{2} \right) = \frac{2\left( \frac{5x - 3}{2} \right) + 3}{5}
   \]

   Simplify the numerator:

   \[
   = \frac{(5x - 3) + 3}{5} = \frac{5x}{5} = x
   \]

   This shows that \( f(f^{-1}(x)) = x \).

#### 2. **Verify \( f^{-1}(f(x)) \):**

   Now substitute \( f(x) \) into \( f^{-1}(x) \).

   - \( f^{-1}(x) = \frac{5x - 3}{2} \), and \( f(x) = \frac{2x + 3}{5} \).

   Now calculate:

   \[
   f^{-1}(f(x)) = f^{-1}\left( \frac{2x + 3}{5} \right) = \frac{5\left( \frac{2x + 3}{5} \right) - 3}{2}
   \]

   Simplify:

   \[
   = \frac{2x + 3 - 3}{2} = \frac{2x}{2} = x
   \]

   This shows that \( f^{-1}(f(x)) = x \).

Both compositions return \( x \), confirming that the functions are inverses of each other.

---

### Summary of Steps:

1. **Write the function as \( y = f(x) \)**.
2. **Switch \( x \) and \( y \)** in the equation.
3. **Solve for \( y \)** to find \( f^{-1}(x) \).
4. **Check by computing both compositions**: \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

### Relative Questions:

1. What is the significance of switching \( x \) and \( y \) in finding an inverse function?
2. Can all functions have an inverse? If not, which ones cannot?
3. How does the domain of the original function relate to the range of the inverse?
4. What happens if the inverse function is not a one-to-one function?
5. Can you explain the geometric interpretation of inverse functions?

### Tip:
Always check if the original function is **one-to-one** (each \( x \) maps to a unique \( y \)) before finding the inverse; otherwise, it may not have an inverse.

Hey can you give me a step by step solution on how to calculate the inverse function in fractional form, and please include the formula on how to check that one.

Learn how to calculate the inverse of a function expressed in fractional form with this step-by-step guide. Understand the process of finding the inverse, verifying it using function composition, and ensuring the function is one-to-one. Suitable for students in Grades 10-12.

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