Inverse Function Notes

Definition of a Function

• A function is a rule that assigns each input to exactly one output.

• Example: $f(x) = x^2$

  • For every input $x$, there is exactly one output $x^2$.

  • If $x = 3$, then $f(3) = 3^2 = 9$.

Vertical Line Test

• A visual way to check if a graph represents a function.

• If any vertical line intersects the graph more than once, the graph is not a function.

• Example: Parabola $y = x^2$ passes the vertical line test.

  • Any vertical line touches the parabola at only one point.

Definition of a One-to-One Function

• For a one-to-one function, each output is the result of exactly one input.

• No different inputs yield the same output.

• Example: $f(x) = x^2$ is not a one-to-one function.

  • For $x = 2$ and $x = -2$, $f(2) = f(-2) = 4$.

  • Two different inputs yield the same output (4).

Horizontal Line Test

• A visual way to check if a function is one-to-one.

• If any horizontal line intersects the graph of the function at more than one point, the function is not one-to-one.

• Example: Parabola $y = x^2$ fails the horizontal line test.

  • A horizontal line like $y = 4$ intersects the parabola at $x = 2$ and $x = -2$.

Examples

• $f(x) = x^3$

  • Passes the vertical line test, so it is a function.

  • Passes the horizontal line test, so it is a one-to-one function.

• $f(x) = \sin(x)$

  • Passes the vertical line test, so it is a function.

  • Fails the horizontal line test, so it is not a one-to-one function.

• $f(x) = e^x$

  • Passes the vertical line test, so it is a function.

  • Passes the horizontal line test, so it is a one-to-one function.

• Circle with radius 2 (

  • Fails the vertical line test, so it is not a function.

  • Fails the horizontal line test, so it is not a one-to-one.

Inverse Functions

• If $f$ is a function from domain $X$ to range $Y$, then its inverse is denoted as $f^{-1}$.

• $f^{-1}$ is a relation from $Y$ to $X$.

• Domain of $f^{-1}$ is the range of $f$, and range of $f^{-1}$ is the domain of $f$.

• $f^{-1}$ acts like an undo button that reverses the mapping of $f$.

• If $f(3) = 8$, then $f^{-1}(8) = 3$ (switching input and output).

Condition for $f^{-1}$ to be a Function

• For $f^{-1}$ to be a function (not just a relation), $f$ must be a one-to-one function.

• If $f$ is not one-to-one, $f^{-1}$ is only a relation.

• Example: $f(x) = x^2$

  • Is a function, but not a one-to-one function.

  • $f^{-1}$ is not a function.

  • $f^{-1}$ takes 4 and gives back two different outputs, 2 and -2, violating the definition of a function.

Important Notes About Inverse Functions

• For every $x$ in the domain of $f$, $f^{-1}(f(x)) = x$.

• For every $y$ in the domain of $f^{-1}$, $f(f^{-1}(y)) = y$.

Steps to Find the Inverse Function

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

2. Switch $x$ with $y$.

3. Solve for $y$.

4. Replace $y$ with $f^{-1}(x)$.

Example 1

• Given: $f(x) = \frac{1}{\sqrt[3]{x - 3}} - 7$

• Find: $f^{-1}(x)$

1. Replace $f(x)$ with $y$: $y = \frac{1}{\sqrt[3]{x - 3}} - 7$

2. Switch $x$ with $y$: $x = \frac{1}{\sqrt[3]{y - 3}} - 7$

3. Solve for $y$:

   • $x + 7 = \frac{1}{\sqrt[3]{y - 3}}$

   • $\sqrt[3]{y - 3} = \frac{1}{x + 7}$

   • $y - 3 = \left(\frac{1}{x + 7}\right)^3$

   • $y = \left(\frac{1}{x + 7}\right)^3 + 3$

4. Replace $y$ with $f^{-1}(x)$: $f^{-1}(x) = \left(\frac{1}{x + 7}\right)^3 + 3$

Example 2

• Given: $f(x) = \frac{3x - 7}{4x + 3}$

• Find: $f^{-1}(x)$

1. Replace $f(x)$ with $y$: $y = \frac{3x - 7}{4x + 3}$

2. Switch $x$ with $y$: $x = \frac{3y - 7}{4y + 3}$

3. Solve for $y$:

   • $x(4y + 3) = 3y - 7$

   • $4xy + 3x = 3y - 7$

   • $4xy - 3y = -3x - 7$

   • $y(4x - 3) = -3x - 7$

   • $y = \frac{-3x - 7}{4x - 3}$

   • $y = \frac{3x + 7}{3 - 4x}$

4. Replace $y$ with $f^{-1}(x)$: $f^{-1}(x) = \frac{3x + 7}{3 - 4x}$

Checking the Work

• Choose an input $x$ for the original function $f$.

• Calculate the output $f(x)$.

• For the inverse function $f^{-1}$, if you input $f(x)$, the output must be $x$.

• Ex: $f(0) = \frac{3(0) - 7}{4(0) + 3} = -\frac{7}{3}$

• $f^{-1}(-\frac{7}{3}) = \frac{3(-\frac{7}{3}) + 7}{3 - 4(-\frac{7}{3})} = 0$

Example 3

• Given: $f(x) = \frac{x^2 + 3}{3x^2}$

• Find: $f^{-1}(x)$

• Problem: Every $x$ variable has an even exponent.

• If we plug in $x$ or $-x$, we will get the same output.

• This function $f$ is not one-to-one; It's a many-to-one function.

• For this original function, which is not one-to-one, you can't find an inverse function.

Even function definition

• Definition $f(x) = f(-x)$

• All even function are symmetric about y-axis, therefore it does not pass the horizontal line test.

How to make it a one-to-one function

• We can restrict its domain to have one-to-one part of it

• If we restrict just ourself to positive part of its domain, under the condition x > 0, you make this function a one-to-one

1. $y = \frac{x^2 + 3}{3x^2}$

2. $x = \frac{y^2 + 3}{3y^2}$

3. Solve of y:

   • $3xy^2 = y^2 + 3$

   • $3xy^2 - y^2 = 3$

   • $y^2(3x - 1) = 3$

   • $y^2 = \frac{3}{3x - 1}$

   • $y = \sqrt{\frac{3}{3x - 1}}$

4. $f^{-1}(x) = \sqrt{\frac{3}{3x - 1}}$

Example 4

• Finding the inverse function of $f(x) = \ln(x)$

1. Change $f(x)$ into $y$: $y = \ln(x)$

2. Switch $x$ and $y$: $x = \ln(y)$

3. Solve for $y$:

   • $x = \log_{e}(y)$

   • $e^{x} = y$

4. Change $y$: $f^{-1}(x) = e^{x}$

Recalling important notes in form of ln and exponential function

• $f(f^{-1}(x)) = x$

• $f^{-1}(f(x)) = x$

• $ln(e^{x}) = x$

• $e^{ln(x)} = x$

Example 5

• Finding the inverse function of $f(x) = log_{2}(x-3) - 5$

1. Change $f(x)$ into $y$: $y = log_{2}(x-3) - 5$

2. Switch $x$ and $y$: $x = log_{2}(y-3) - 5$

3. Solve for $y$:

   • $x = log_{2}(y-3) - 5$

   • $x + 5 = log_{2}(y-3)$

   • $2^{x+5} = y-3$

   • $2^{x+5} + 3 = y$

4. Change $y$: $f^{-1}(x) = 2^{x+5} + 3$