Analyzing relations and functions

Published on February 6, 2025

Functions

The provided text describes functions, relations, domains, ranges, and inverse functions. It includes examples of relations that are not functions, and explains how to determine the domain and range of a function, including a quadratic function and a rational function. It also demonstrates how to find the inverse of a function.

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Here are the step-by-step solutions to the math problems:

i. Describe a function in terms of relations, input and output.

A function is a special type of relation between two sets, say set A (domain) and set B (codomain), such that each element in set A is associated with exactly one element in set B.

In terms of relations, a function is a set of ordered pairs

where no two distinct ordered pairs have the same first element

Input refers to the element from the domain (set A) that is given to the function. It is the value that is used as an argument for the function, often denoted by

Output refers to the element from the codomain (set B) that the function associates with a given input. It is the value that the function produces, often denoted by

. For each input, a function produces exactly one output.

ii. Which of the following relations is a function? Find the domain, type of relation and the range of the function.

α. If , where , and .

Is a function?
To check if is a function, we look for repeated first elements in the ordered pairs. We see that the first element ‘2’ appears in two ordered pairs: and . Since the input 2 is associated with two different outputs (2 and 1), is not a function.
Domain of :
The domain of is the set of all first elements in the ordered pairs of .
Range of :
The range of is the set of all second elements in the ordered pairs of .
Type of relation:
is a relation but not a function.

β. If , where , and .

Is a function?
Similar to relation , we check for repeated first elements in the ordered pairs of . We see that the first element ‘2’ appears in two ordered pairs: and . Since the input 2 is associated with two different outputs (2 and 1), is not a function.
Domain of :
The domain of is the set of all first elements in the ordered pairs of .
Range of :
The range of is the set of all second elements in the ordered pairs of .
Type of relation:
is a relation but not a function.

iii. Sketch the graph of the function and describe the type of function.

The function is

.
This is a quadratic function. The graph is a parabola.
Since the coefficient of

is positive (2 > 0), the parabola opens upwards.
The vertex of the parabola is at

because the standard form of a parabola is

, where

is the vertex. Here

, so

and

To sketch the graph, we can find a few points:
For

. Point:

(vertex)
For

. Point:

For

. Point:

For

. Point:

For

. Point:

The graph is a U-shaped parabola opening upwards with vertex at

Type of function:

Quadratic function: because it is a polynomial of degree 2.
Polynomial function: because it is defined by a polynomial expression.
Even function: because .

iv. Given that , state the domain of .

For the function

to be defined in the real numbers, the expression under the square root must be non-negative. Also, the denominator

cannot be zero.
Therefore, the domain of

is the set of all

such that:

and

This means that either both

and

are positive, or both

and

are negative, or

and

. In summary, we need to solve the inequality

, keeping in mind that

v. The function is defined on the set of real numbers by , where . Find the inverse of the function .

Let

. To find the inverse function

, we need to solve for

in terms of

Multiply both sides by

Rearrange the terms to solve for

Factor out

from the left side:

Divide by

to isolate

So, the inverse function is

.
Replacing

with

to express the inverse function in terms of

The inverse function is

Final Answer: The final answer is

α β

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Analyzing relations and functions

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