Anasayfa/ AP Calculus/ Inverse Functions
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πŸ”„ Inverse Functions

One-to-one functions, horizontal line test, finding inverses, and $f^{-1}(x)$
🎯 In this section you will learn

πŸ“Œ One-to-One Functions

A function $f$ is one-to-one if each output corresponds to exactly one input. In other words, $f(a) = f(b)$ implies $a = b$.

πŸ“ Horizontal Line Test
A function has an inverse if and only if no horizontal line intersects its graph more than once.
FunctionOne-to-One?Reason
$f(x) = x^2$No$f(-2) = f(2) = 4$
$f(x) = x^3$YesStrictly increasing
$f(x) = \sin x$NoPeriodic: repeats values
$f(x) = e^x$YesStrictly increasing

πŸ“ Definition of Inverse Function

If $f$ is one-to-one, its inverse $f^{-1}$ satisfies:

$$ f^{-1}(f(x)) = x \quad \text{for all } x \text{ in the domain of } f $$ $$ f(f^{-1}(x)) = x \quad \text{for all } x \text{ in the domain of } f^{-1} $$
πŸ’‘ Important: Not $1/f(x)$
The notation $f^{-1}$ means "inverse function", not "reciprocal". $\displaystyle f^{-1}(x) \neq \frac{1}{f(x)}$.

πŸ”§ Finding the Inverse of a Function

β‘ 
Replace $f(x)$ with $y$
Write the function as $y = f(x)$.
β‘‘
Swap $x$ and $y$
Interchange $x$ and $y$ to get $x = f(y)$.
β‘’
Solve for $y$
Isolate $y$ to express it in terms of $x$.
β‘£
Replace $y$ with $f^{-1}(x)$
Write the result as $f^{-1}(x)$.
Example 1Finding an Inverse (Linear)

Find the inverse of $f(x) = 2x + 3$.

β‘ 
$y = 2x + 3$
Replace $f(x)$ with $y$.
β‘‘
$x = 2y + 3$
Swap $x$ and $y$.
β‘’
Solve for $y$
$x - 3 = 2y \Rightarrow y = \frac{x-3}{2}$
β‘£
$f^{-1}(x) = \frac{x-3}{2}$
Done.
Example 2Finding an Inverse (Quadratic with Restriction)

Find the inverse of $f(x) = x^2$ for $x \ge 0$.

β‘ 
$y = x^2$, $x \ge 0$
Original function with domain restriction.
β‘‘
$x = y^2$, $y \ge 0$
Swap $x$ and $y$. Note $y \ge 0$ from original domain.
β‘’
$y = \sqrt{x}$
Solve for $y$, taking the positive root (since $y \ge 0$).
β‘£
$f^{-1}(x) = \sqrt{x}$
Domain of $f^{-1}$ is $[0, \infty)$.

πŸ“Š Graph of an Inverse Function

The graph of $f^{-1}$ is the reflection of the graph of $f$ across the line $y = x$.

Example 3Graphing $f$ and $f^{-1}$

For $f(x) = 2x + 3$ and $f^{-1}(x) = \frac{x-3}{2}$, the graphs are symmetric about $y = x$.

πŸ–ΌοΈ Visual Demonstration

Interactive graph below. Select a function and see its inverse reflected across $y = x$:

πŸ” Key Takeaway
Inverse functions "undo" each other. They are essential for solving equations and appear frequently in calculus (inverse trig functions, logarithms, etc.).

πŸ“ Practice Problems

Problem 1Find the Inverse

Find the inverse of $f(x) = 4x - 7$.

Show solution

$y = 4x - 7 \Rightarrow x = 4y - 7 \Rightarrow 4y = x + 7 \Rightarrow y = \frac{x+7}{4}$. So $f^{-1}(x) = \frac{x+7}{4}$.

Problem 2One-to-One?

Is $f(x) = |x|$ one-to-one? Why or why not?

Show solution

No. $f(-2) = 2$ and $f(2) = 2$, so different inputs give the same output.

Problem 3Inverse with Restriction

Find the inverse of $f(x) = x^2 + 1$ for $x \ge 0$.

Show solution

$y = x^2 + 1$, $x \ge 0$ β‡’ $x = y^2 + 1$, $y \ge 0$ β‡’ $y^2 = x - 1$ β‡’ $y = \sqrt{x-1}$. So $f^{-1}(x) = \sqrt{x-1}$, domain $[1, \infty)$.

Problem 4Composition Check

Verify that $f(x) = 3x + 2$ and $g(x) = \frac{x-2}{3}$ are inverses.

Show solution

$f(g(x)) = 3\left(\frac{x-2}{3}\right) + 2 = x - 2 + 2 = x$. $g(f(x)) = \frac{(3x+2)-2}{3} = \frac{3x}{3} = x$. So yes, they are inverses.

Problem 5Domain of Inverse

If $f(x) = \sqrt{x-2}$, what are the domain and range of $f^{-1}$?

Show solution

Domain of $f$: $[2, \infty)$, range of $f$: $[0, \infty)$. For $f^{-1}$, domain = range of $f$ = $[0, \infty)$, range = domain of $f$ = $[2, \infty)$.

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