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