Introduction to Inverse Functions: Videos & Practice Problems

Understanding one-to-one functions is essential, where each output pairs with a unique input, verified by the horizontal line test on graphs. Such functions have inverse functions, denoted as $f^{-1}$, which swap inputs and outputs, reversing the domain and range. This concept is crucial for analyzing relations and their inverses, enhancing comprehension of functions, ordered pairs, and mappings. Recognizing these properties supports deeper insights into polynomial behavior, exponents, and function transformations, foundational for mastering algebraic structures and their applications.

One to One Functions

Understanding the concept of a one-to-one function builds upon the foundational idea of a function, where each input (x-value) corresponds to at most one output (y-value). A one-to-one function, however, requires a stricter condition: each output must be paired with at most one input. This means no two different inputs share the same output value.

To visualize this, consider a set of ordered pairs representing a function. If any output value repeats for different inputs, the function is not one-to-one. For example, if the output 2 corresponds to both inputs -4 and 1, this violates the one-to-one condition. This can be quickly identified by checking for repeated y-values in the list of ordered pairs or by observing a correspondence diagram where multiple arrows point to the same output.

When analyzing functions graphically, the vertical line test helps confirm whether a graph represents a function by ensuring no vertical line intersects the graph at more than one point. To determine if a function is one-to-one, the horizontal line test is used. This test states that if any horizontal line intersects the graph at more than one point, the function is not one-to-one. Passing the horizontal line test confirms that each output corresponds to a unique input, reinforcing the one-to-one nature of the function.

In summary, a one-to-one function is characterized by a unique pairing between inputs and outputs, where no output is shared by multiple inputs. This can be verified through ordered pairs by checking for repeated outputs, through correspondence diagrams by observing arrow mappings, or through graphs by applying the horizontal line test. Recognizing one-to-one functions is essential for understanding inverse functions and many other advanced mathematical concepts.

One to One Functions Example 1

To determine whether a graph represents a one-to-one function, it is essential to first verify if the graph is a function by applying the vertical line test. The vertical line test states that if any vertical line intersects the graph at more than one point, the graph does not represent a function. Once confirmed as a function, the horizontal line test is used to check if the function is one-to-one. A one-to-one function passes the horizontal line test, meaning no horizontal line intersects the graph at more than one point.

For example, consider a parabola. Applying the vertical line test shows that no vertical line intersects the parabola at more than one point, confirming it is a function. However, when applying the horizontal line test, some horizontal lines intersect the parabola at two points, indicating it is not a one-to-one function.

Next, consider a linear graph. The vertical line test confirms it is a function since no vertical line intersects it more than once. The horizontal line test also confirms it is one-to-one because no horizontal line intersects the graph at more than one point. Therefore, this linear graph represents a one-to-one function.

Finally, for another graph, applying the vertical line test confirms it is a function, and the horizontal line test shows it is one-to-one as well, since no horizontal line intersects the graph more than once.

Understanding and applying the vertical and horizontal line tests are fundamental for identifying functions and determining if they are one-to-one. This knowledge is crucial for further exploration of function properties and their inverses.

Which of the following is the graph of a one-to-one function?

• A.
• B.
• C.
• D.

Answer: (A)

Which of the following is the graph of a one-to-one function?

• A.
• B.
• C.
• D.

Answer: (D)

Consider the following relations. Which is a one-to-one function?

• A. $\left\lbrace{(1,4),(2,5),(3,6),(4,7)}\right\rbrace$
• B. $\left\lbrace{(1,2),(1,3),(2,4)}\right\rbrace$
• C. ${\left\lbrace(2,3),(3,3),(4,5)\right\rbrace}$
• D. ${\left\lbrace(1,1),(2,2),(3,1),(9,9)\right\rbrace}$

Answer: (A)

Intro to Inverse Functions

When a function is one-to-one, it possesses a unique and important feature known as an inverse function. An inverse function essentially reverses the roles of inputs and outputs from the original function. If the original function is denoted as f, its inverse is written as f⁻¹, where the superscript ⁻¹ indicates the inverse, not an exponent or reciprocal.

For a one-to-one function f with ordered pairs (x, y), the inverse function f⁻¹ consists of ordered pairs (y, x), effectively swapping each input and output. This means that if f maps x to y, then f⁻¹ maps y back to x.

For example, consider the function f defined by the ordered pairs (−4, 2), (−2, −1), and (1, 3). Since each output corresponds to exactly one input, f is one-to-one and has an inverse. The inverse function f⁻¹ swaps these pairs to (2, −4), (−1, −2), and (3, 1).

This swapping also applies to the domain and range of the functions. The domain of the original function f (the set of all x-values) becomes the range of the inverse function f⁻¹, and the range of f (the set of all y-values) becomes the domain of f⁻¹. Understanding this relationship is crucial for working with inverse functions and solving problems involving them.

Intro to Inverse Functions Example 2

When determining whether a relation represented by a table of values is a function, the key criterion is that each input corresponds to at most one output. If any input maps to multiple outputs, the relation is not a function. For example, if the input 1 maps to both 4 and 7, this violates the definition of a function. However, if every input has a unique output, the relation qualifies as a function.

To assess if a function is one-to-one (injective), each output must correspond to at most one input. This means no two different inputs share the same output value. For instance, if both inputs 1 and 2 map to the output 4, the function is not one-to-one. When a function is one-to-one, it guarantees the existence of an inverse function, which essentially reverses the mapping of inputs and outputs.

The inverse function, denoted as f⁻¹(x), is found by swapping the inputs and outputs of the original function f(x). For example, if the original function maps inputs 1, 2, 3, and 4 to outputs 4, 7, 9, and 12 respectively, the inverse function will map inputs 4, 7, 9, and 12 back to outputs 1, 2, 3, and 4. This reversal is fundamental in understanding inverse functions and their properties.

It is important to note that only one-to-one functions have inverses that are also functions. If a function is not one-to-one, such as when two different inputs share the same output, the inverse relation will not pass the vertical line test and thus is not a function. Consequently, the inverse function cannot be defined in these cases.

Consider the set of ordered pairs. Verify if it is one-to-one. If so, find its inverse.
$f=\left\{\right(2,9),(4,11),(6,15),(8,20\left)\right\}$

• A. $f^{-1}=\left\{\right(\frac{1}{9},\frac{1}{2}),(\frac{1}{11},\frac{1}{4}),(\frac{1}{15},\frac{1}{6}),(\frac{1}{20},\frac{1}{8}\left)\right\}$
• B. $f^{-1}=\left\{\right(\frac{1}{2},\frac{1}{9}),(\frac{1}{4},\frac{1}{11}),(\frac{1}{6},\frac{1}{15}),(\frac{1}{8},\frac{1}{20}\left)\right\}$
• C. $f^{-1}=\left\{\right(9,2),(11,4),(15,6),(20,8\left)\right\}$
• D. The function is not one-to-one.

Answer: (C)

Consider the set of ordered pairs. Verify if it is one-to-one. If so, find its inverse.
$g=\left\{\right(1,\frac{3}{2}),(4,\frac{5}{3}),(7,\frac{9}{4}),(10,\frac{11}{6}\left)\right\}$

• A. $g^{-1}={\left\lbrace(1,\frac23),\left(\frac14,\frac35\right),(\frac17,\frac49),(\frac{1}{10},\frac{6}{11})\right\rbrace}$
• B. $g^{-1}=\left\{\right(\frac{2}{3},1),(\frac{3}{5},\frac{1}{4}),(\frac{4}{9},\frac{1}{7}),(\frac{6}{11},\frac{1}{10}\left)\right\}$
• C. $g^{-1}=\left\{\right(\frac{3}{2},1),(\frac{5}{3},4),(\frac{9}{4},7),(\frac{11}{6},10\left)\right\}$
• D. The function is not one-to-one.

Answer: (C)

Here’s what students ask on this topic:

What is a one-to-one function and how can I determine if a function is one-to-one?

A one-to-one function is a function where each output (y-value) is paired with at most one input (x-value). This means no two different inputs share the same output. To determine if a function is one-to-one, you can use the horizontal line test on its graph: if any horizontal line intersects the graph at more than one point, the function is not one-to-one. Alternatively, when given ordered pairs or a correspondence diagram, check if any output value repeats for different inputs. If it does, the function is not one-to-one. Understanding this concept is important because only one-to-one functions have inverse functions.

How do I find the inverse of a one-to-one function?

To find the inverse of a one-to-one function $f$, you swap the inputs and outputs of the function. If the original function has ordered pairs $(x,y)$, the inverse function $f^{-1}$ will have ordered pairs $(y,x)$. This means the domain and range switch places. For example, if $f$ maps $x$ to $y$, then $f^{-1}$ maps $y$ back to $x$. Note that the notation $f^{-1}$ does not mean an exponent but denotes the inverse function. This process only works if $f$ is one-to-one.

What is the horizontal line test and why is it important?

The horizontal line test is a graphical method used to determine if a function is one-to-one. To perform the test, draw horizontal lines across the graph of the function. If any horizontal line intersects the graph at more than one point, the function is not one-to-one. This is important because only one-to-one functions have inverse functions. The test helps quickly identify whether each output corresponds to a unique input, which is essential for understanding the behavior of functions and their inverses.

How do the domain and range change when finding the inverse of a function?

When finding the inverse of a function, the domain and range swap roles. The domain of the original function becomes the range of the inverse function, and the range of the original function becomes the domain of the inverse. This happens because the inverse function reverses the input-output pairs. For example, if the original function $f$ has domain $D$ and range $R$, then its inverse $f^{-1}$ has domain $R$ and range $D$. Understanding this swap is crucial when working with inverse functions and their graphs.

Why is it important for a function to be one-to-one to have an inverse?

A function must be one-to-one to have an inverse because the inverse function reverses the roles of inputs and outputs. If a function is not one-to-one, it means some outputs correspond to multiple inputs, so swapping inputs and outputs would not produce a well-defined function. In other words, the inverse would assign a single input to multiple outputs, which violates the definition of a function. Therefore, only one-to-one functions have inverses that are also functions, ensuring each input in the inverse corresponds to exactly one output.