Inverse Relations in Graphs

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Questions and Answers

What defines two functions f and g as being inverses of each other?

g(f(x))=x and f(g(x))=x for all x in the domain of f
g(f(x))=f(g(x)) for every x in the domain of f
f(g(x))=x and g(f(x))=x for every x in the domain of g (correct)
f(g(x))=g(f(x)) for all x in the domain of f

Which statement accurately describes a one-to-one function?

There can be repeated outputs for different inputs in the domain.
If f(x1) = f(x2), then x1 may or may not equal x2.
Each output in the range corresponds to exactly one input in the domain. (correct)
Every element in the domain maps to multiple elements in the range.

What does the horizontal line test determine about a function?

If the function is one-to-one. (correct)
If the function is even or odd.
If the function intersects the y-axis at multiple points.
If the function is defined over multiple intervals.

Why is the function y = x² - 4x + 7 not considered one-to-one on the real numbers?

It outputs the same value for two different inputs. (D) Signup and view all the answers

What is the significance of every element f(x) in the range of a one-to-one function?

It must originate from only one element in the domain. (A) Signup and view all the answers

What is a characteristic of a one-to-one function?

Each output corresponds to exactly one input. (C) Signup and view all the answers

Which of the following functions is an example of a one-to-one function?

$f(x) = 3x + 2$ (D) Signup and view all the answers

What is the first step to determine if a function has an inverse?

Use the horizontal line test. (D) Signup and view all the answers

What represents the domain of a function?

The values of x for which the function is defined. (C) Signup and view all the answers

What process is NOT involved in finding the inverse of a one-to-one function?

Graph the inverse function. (D) Signup and view all the answers

Which of the following statements is true regarding inverse functions?

If a function is one-to-one, it has an inverse function. (B) Signup and view all the answers

If a function is represented as y = 5x + 3, what would be the first step to find its inverse?

Swap x and y. (C) Signup and view all the answers

Why is the circumference function for a circle considered a one-to-one function?

It has a unique output for each input diameter. (C) Signup and view all the answers

What condition must be satisfied for the inverse relation of a function to also be a function?

The function must be one-to-one. (D) Signup and view all the answers

Which equation represents the inverse function of f(x) = 3x + 2?

y = (x - 2)/3 (B) Signup and view all the answers

What is the result of composing a function with its inverse function?

The original input value. (B) Signup and view all the answers

What does passing the horizontal line test indicate about a function?

The function has a unique output for every input. (D) Signup and view all the answers

Geometrically, how is the graph of the inverse relation derived from the graph of the original function?

By reflecting it across the line y = x. (D) Signup and view all the answers

In the equation x = 3y + 2, which step leads to finding the inverse relationship?

Switch x and y. (A) Signup and view all the answers

For the function y = f(x), what notation commonly represents its inverse function?

f^-1(x) (A) Signup and view all the answers

What is the geometric significance of the line y = x in relation to a function and its inverse?

It serves as the axis of symmetry. (C) Signup and view all the answers

Which function is one-to-one?

y = x^3 (D) Signup and view all the answers

What is the range of the function y = |x| + 1?

[1, ∞) (A) Signup and view all the answers

What would be the ordered pairs of the inverse relation for the function y = |x| + 1?

{(3, -2), (2, -1), (1, 0)} (A) Signup and view all the answers

What is the domain of the function y = |x| + 1?

All real numbers (D) Signup and view all the answers

Which statement is true regarding the function and its inverse relation?

The domain of the inverse is the range of the function. (D) Signup and view all the answers

What does it mean for a relation to not be a function?

It has varying outputs for the same input. (C) Signup and view all the answers

How do the graphs of a function and its inverse relate to the line y = x?

They are reflections across the line y = x. (B) Signup and view all the answers

What is one characteristic of the inverse relation of f = {(1, 1), (2, 3), (3, 1), (4, 2)}?

It includes duplicates in the output. (B) Signup and view all the answers

Study Notes

Inverse Relations and Functions

Reflect the graph of a function in the line y = x to obtain its inverse.
Ordered pairs of a function ( f(x) ) can be transformed into ordered pairs of its inverse by switching inputs and outputs.
The inverse of a function ( y = f(x) ) is denoted as ( y = f^{-1}(x) ).

Conditions for Inverse Relation to be a Function

An inverse function exists only if the original function ( f ) is a one-to-one function.
A function is one-to-one if no two different inputs produce the same output.
The horizontal line test determines if a function is one-to-one; if a horizontal line intersects the graph at more than one point, the function is not one-to-one.

Finding the Inverse Function Algebraically

Start with the function’s equation: replace ( f(x) ) with ( y ).
Interchange ( x ) and ( y ) to seek the inverse.
Solve the resulting equation for ( y ) and replace ( y ) with ( f^{-1}(x) ).

Concepts and Definitions

Domain: The set of all possible ( x ) values for which the function is defined.
Range: The set of all resulting ( y ) values derived from the domain values.
One-to-One Function: A function where ( f(a) = f(b) ) implies ( a = b ).

Examples of One-to-One Functions

Circumference of a Circle: Given by ( C = \pi d ), it's one-to-one since each diameter yields a unique circumference.
Pricing Function: For a price of apples at Php 28.00 each, the total price function ( P = 28x ) is one-to-one, as each quantity ( x ) corresponds to a distinct total price.

Existence of an Inverse Function

A function ( f ) has an inverse if it passes the horizontal line test.
The inverse function ( g ) satisfies ( f(g(x)) = x ) and ( g(f(x)) = x ) for all ( x ) in the respective domains.

Horizontal Line Test

A function is one-to-one if a horizontal line intersects its graph at most once.
For example, the quadratic function ( y = x^2 - 4x + 7 ) fails the horizontal line test as it intersects ( y = 7 ) twice.

Ordered Pairs and Inverses

Ordered pairs of the function ( f ) are reversed for the corresponding inverse relation.
Example: For ( f = {(1, 1), (2, 3), (3, 1), (4, 2)} ), the inverse relation is ( f^{-1} = {(1, 1), (3, 2), (1, 3), (2, 4)} ).
The domain of the inverse is the range of the original function, and vice versa.

Summary

Reflecting a function across the line ( y=x ) creates its inverse.
Only one-to-one functions have inverses, confirming the significance of the horizontal line test.
Understanding the relationship between the function and its inverse aids in grasping core principles of mathematical functions.

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Description

This quiz explores how to find the graph of an inverse relation from the graph of a given function. You will analyze the ordered pairs and their transformations to understand the relationship between a function and its inverse geometrically. Dive into the concepts of symmetry and mapping on the Cartesian plane.