Describe the steps involved in finding the inverse of a function algebraically.

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To find the **inverse** of a function algebraically, you essentially want to reverse the roles of the input and the output. The inverse function, often denoted as $$ f^{-1}(x) $$, satisfies the condition that $$ f^{-1}(f(x)) = x $$ and $$ f(f^{-1}(x)) = x $$ for all $$ x $$ in the domain of $$ f $$ and $$ f^{-1} $$, respectively.

Here are the step-by-step procedures to find the inverse of a function algebraically:

### **1. Verify that the Function Has an Inverse**

Before attempting to find an inverse, ensure that the function is **bijective** (both **one-to-one** and **onto**):

- **One-to-One (Injective):** Each element of the function's domain maps to a unique element in its codomain.
- **Onto (Surjective):** Every element in the function's codomain is mapped by some element in its domain.

A function that is both injective and surjective is **bijective**, which guarantees the existence of an inverse function.

*Example:*  
Consider $$ f(x) = 2x + 3 $$. This is a linear function with a slope of 2 and a y-intercept of 3. Since it's a straight line with a non-zero slope, it is one-to-one and onto, hence it has an inverse.

### **2. Replace $$ f(x) $$ with $$ y $$**

Start by rewriting the function using $$ y $$ to make the equations easier to manipulate.

$$ y = f(x) $$

*Example:*  
$$ y = 2x + 3 $$

### **3. Swap $$ x $$ and $$ y $$**

Interchange the roles of $$ x $$ and $$ y $$. This step is based on the idea that the inverse function reverses the original function's operation.

$$ x = f(y) $$

*Example:*  
$$ x = 2y + 3 $$

### **4. Solve for $$ y $$**

Algebraically manipulate the equation to solve for $$ y $$. This $$ y $$ will represent the inverse function $$ f^{-1}(x) $$.

*Example:*
$$ x = 2y + 3 $$
Subtract 3 from both sides:
$$ x - 3 = 2y $$
Divide both sides by 2:
$$ y = \frac{x - 3}{2} $$

### **5. Replace $$ y $$ with $$ f^{-1}(x) $$**

Once you've solved for $$ y $$, denote this expression as the inverse function.

$$ f^{-1}(x) = \frac{x - 3}{2} $$

### **6. (Optional) Verify the Inverse Function**

To ensure that you've correctly found the inverse, you can verify it by composing the original function and its inverse:

- $$ f(f^{-1}(x)) = x $$
- $$ f^{-1}(f(x)) = x $$

*Verification:*
1. $$ f(f^{-1}(x)) = f\left(\frac{x - 3}{2}\right) = 2\left(\frac{x - 3}{2}\right) + 3 = x - 3 + 3 = x $$
2. $$ f^{-1}(f(x)) = f^{-1}(2x + 3) = \frac{(2x + 3) - 3}{2} = \frac{2x}{2} = x $$

Since both compositions return $$ x $$, the inverse function is correctly determined.

### **Summary of Steps**

1. **Verify** the function is bijective.
2. **Express** the function as $$ y = f(x) $$.
3. **Swap** $$ x $$ and $$ y $$ to get $$ x = f(y) $$.
4. **Solve** the resulting equation for $$ y $$.
5. **Rename** $$ y $$ as $$ f^{-1}(x) $$.
6. **(Optional) Verify** by composition.

### **Additional Considerations**

- **Domain and Range:** The domain of $$ f $$ becomes the range of $$ f^{-1} $$, and vice versa. Ensure that the inverse function's domain and range are correctly identified.
  
- **Horizontal Line Test:** To confirm that a function is one-to-one (and thus has an inverse), you can use the Horizontal Line Test. If any horizontal line intersects the graph of the function more than once, the function does not have an inverse.

- **Restricted Domains:** Sometimes, a function may not be one-to-one over its entire domain but can be made one-to-one by restricting its domain. This is common with functions like $$ f(x) = x^2 $$, which does not have an inverse unless its domain is limited to $$ x \geq 0 $$ or $$ x \leq 0 $$.

### **Another Example**

**Find the inverse of** $$ f(x) = \frac{3x - 5}{2} $$.

1. **Express as $$ y $$:**
   $$ y = \frac{3x - 5}{2} $$

2. **Swap $$ x $$ and $$ y $$:**
   $$ x = \frac{3y - 5}{2} $$

3. **Solve for $$ y $$:**
   $$ 2x = 3y - 5 $$
   $$ 2x + 5 = 3y $$
   $$ y = \frac{2x + 5}{3} $$

4. **Write the inverse function:**
   $$ f^{-1}(x) = \frac{2x + 5}{3} $$

5. **Verify:**
   $$ f(f^{-1}(x)) = \frac{3\left(\frac{2x + 5}{3}\right) - 5}{2} = \frac{2x + 5 - 5}{2} = \frac{2x}{2} = x $$
   $$ f^{-1}(f(x)) = \frac{2\left(\frac{3x - 5}{2}\right) + 5}{3} = \frac{3x - 5 + 5}{3} = \frac{3x}{3} = x $$

Since both compositions yield $$ x $$, the inverse is correctly found.

By following these systematic steps, you can find the inverse of most algebraic functions, provided they are one-to-one and thus have an inverse.
To find the inverse of a function algebraically, follow these steps: 1. **Verify** the function is one-to-one. 2. **Express** the function as $$ y = f(x) $$. 3. **Swap** $$ x $$ and $$ y $$ to get $$ x = f(y) $$. 4. **Solve** for $$ y $$. 5. **Rename** $$ y $$ as $$ f^{-1}(x) $$. 6. **(Optional) Verify** by composition. This process ensures you find the correct inverse function.

Describe the steps involved in finding the inverse of a function algebraically.

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