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Inverse Relations and Functions Solutions

• Recall that a    relation R     is a set of ordered pairs.
• A function is a relation in which there is    only one output    for every input.

Note

In other words, the domain contains a unique set of values.

 

Functions were introduced in Algebra 1: Principles of Secondary Mathematics. This lesson continues to grow this knowledge.

•    Inverses    can be applied to relations and functions.
• When the inverse of a relation (or function) $R^{–1}$ is found, the relationship between the domain and range is    reversed   .
• This means that the coordinates are    interchanged   , or reflected over the line $y=x$ when graphed.
• Two relations R and Q are inverses of one another, if and only if every ordered pair in R is $(a, b)$ and every ordered pair in Q is $(b, a)$ where $\underline{ \mathbf{\{}\mathit{a}\mathbf{,}\mathbf{ }\mathit{b}\mathbf{\in }\mathbf{ℂ}\mathbf{\}} }$

Note

The notation $\left\{a, b\in \mathrm{ℂ}\right\}$ is read “a and b are elements of the set complex numbers.”

 

The phrase “if and only if” in mathematics tells you that the statement is true forward and backward.

 

In mathematics, there are often exceptions to rules. Therefore, take special note when you encounter words like “every,” “always,” and “must.” These words mean there are no exceptions to the rule.

• • The    domain    of R is the    range    of Q.
  • The    range    of R is the    domain    of Q.

Example 1

Represent relation R as a table. Represent the inverse of the relation ${\mathit{R}}^{\mathbf{–}\mathbf{1}}$ as a mapping.

$R=\left\{\left(–2, –3\right), \left(1, –1\right), \left(0, 2\right), \left(3, 5\right)\right\}$

Note

Recall that when mapping, write the domain and range values in numerical order.

Compare the domain and range of R and ${\mathit{R}}^{\mathbf{–}\mathbf{1}}\mathbf{.}$ Explain if either R or ${\mathit{R}}^{\mathbf{–}\mathbf{1}}$ are functions.

The domain of R is the range of $R^{–1}$ and the range of R is  $\underline{ \mathbf{the}\mathbf{ }\mathbf{domain}\mathbf{ }\mathbf{of} {\mathit{R}}^{\mathbf{–}\mathbf{1}} }$.

$\underline{ \mathbf{Both}\mathbf{ }\mathbf{R}\mathbf{ }\mathbf{and} {\mathit{R}}^{\mathbf{–}\mathbf{1}}\mathbf{ }\mathbf{are}\mathbf{ }\mathbf{functions} }$ because each domain is unique.

 

Example 2

A small town surveyed its community to determine the type of transportation townspeople use in a typical week. The survey collected the type of transportation people used and the number of wheels associated with the transportation. A relation was written to compile the data.

Find the inverse of the relation. Explain how you know if the relation and its inverse are functions or not.

W = {(bicycle, 2), (car, 4), (truck, 4), (semi-truck cab, 10), (motorcycle, 2), (feet, 0), (SUV, 4)}

$W^{–1}$

domain: {0, 2, 4, 10}
range: {bicycle, car, feet, motorcycle, semi-truck, SUV, truck}