Inverse functions are a fundamental concept in algebra that allow us to find a function that reverses the effect of another function. This means if we have a function $f(x)$ that takes an input $x$ and produces an output $y$, the inverse function, denoted as $f(x)$, will take $y$ as an input and produce the original $x$as an output. Understanding how to find and use inverse functions is crucial for solving equations and understanding the relationship between variables in various mathematical contexts.

**Understanding Inverse Functions**

To understand inverse functions, one must grasp that the inverse essentially "undoes" the action of the original function. For a function to have an inverse, each input must have a unique output, and each output must come from a unique input. This property is known as being 'one-to-one'. A graphical representation of a function that has an inverse is a curve that passes the horizontal line test - meaning no horizontal line intersects the graph more than once.

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**Characteristics of Inverse Functions**

**One-to-one correspondence:**For $f(x)$ to have an inverse, no two different inputs can map to the same output.**Function notation:**The inverse of $f(x)$ is denoted as $f(x)$, which is read as "f inverse of x".**Domain and Range:**The domain of $f(x)$ becomes the range of $f(x)$, and vice versa.**Graphical relationship:**The graph of $f(x)$ is a reflection of the graph of $f(x)$ across the line $y = x.$

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**Finding Inverse Functions**

To find the inverse of a function, follow these steps:

1. Replace $f(x)$ with $y$.

2. Swap $x$ and $y$.

3. Solve for $y$, which becomes $f(x)$.

**Example 1: Linear Function**

Given $f(x) = 2x + 3$, find $f(x)$.

1. Replace $f(x)$ with $y$: $y = 2x + 3$.

2. Swap $x$ and $y$: $x = 2y + 3$.

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

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

**Example 2: Quadratic Function**

Consider $f(x) = x^2 + 4x + 3$, where $x \geq -2$to ensure the function is one-to-one.

1. Replace $f(x)$ with $y$: $y = x^2 + 4x + 3$.

2. Swap $x$and $y$: $x = y^2 + 4y + 3$.

3. Solve for $y$: This step involves completing the square or using the quadratic formula. After manipulation, you find that $y$ in terms of $x$ corresponds to the inverse function, ensuring to only consider the branch where $x \geq -2$.

**Composite Functions and Inverse Functions**

Understanding composite functions is key to working with inverses. The composition of a function $f$with its inverse $f^{-1}$ will always yield the original input value for $x$, i.e., $f(f^{-1}(x)) = x$and $f^{-1}(f(x)) = x$.

**Example 3: Verifying Inverses**

If $f(x) = 3x - 5$ and $g(x) = \frac{x + 5}{3}$, show that $f$ and $g$ are inverses.

1. Compute $f(g(x))$: $f(g(x)$ = $3\left(\frac{x + 5}{3}\right) - 5 = x$.

2. Compute $g(f(x))$: $g(f(x)$= $\frac{(3x - 5) + 5}{3} = x$.

Since both compositions return $x$, $f$ and $g$ are indeed inverses of each other.

**Practice Questions**

1. Find the inverse function of** **$f(x) = 5x - 7$**.**

2. Given $f(x) = \sqrt{x + 3}$, find $f^{-1}(x)$.

3. If $g(x) = \frac{1}{2x - 4},$ determine $(g^{-1}(x)$.

**Solutions**

1. For $f(x) = 5x - 7$, $f^{-1}(x) = \frac{x + 7}{5}$.

2. For $f(x) = \sqrt{x + 3}$, swapping $x$ and $y$ and solving for $y$ gives $f^{-1}(x) = x^2 - 3$, considering the domain $x \geq 0$.

3. For $g(x) = \frac{1}{2x - 4}$, $g^{-1}x$= $\frac{1 + 4x}{2x}$, ensuring to follow the steps of swapping $x$ and $y$ and solving for$y$.

**Key Takeaways**

- Inverse functions reverse the operation of the original function.
- To find the inverse, swap $x$ and $y$ in the equation and solve for $y$.
- The graph of an inverse function is the reflection of the original function across the line $y = x$.
- Verifying inverses can be done by showing that $f(f^{-1}(x) = x$ and $f^{-1}(f(x)) = x$.