Relations and Functions in Mathematics

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Questions and Answers

The basic properties of the logarithmic function $f(x) = log_b x$ are used in various fields, such as ______ and ______.

acoustics, chemistry

What is the value of $x$ in the equation $log_x(2401) = 4$ ?

  • 2401
  • 7 (correct)
  • 4
  • 16

The decibel (dB) level of a sound is calculated using the formula $D = 10 log_{10}(I/I_0)$ where $I$ is the sound intensity and $I_0$ is the reference intensity.

True (A)

What does the pH level measure in a water-based solution?

<p>The pH level measures the acidity of a water-based solution.</p> Signup and view all the answers

Match the following quantities with their corresponding logarithmic scales:

<p>Sound intensity = Decibel (dB) Earthquake magnitude = Richter scale Acidity of a solution = pH scale</p> Signup and view all the answers

What is the formula for the cost, C(x), representing the total expenses of a function with a fixed cost and a variable cost per attendee?

<p>C(x) = 200x + 12,000 (B)</p> Signup and view all the answers

In the given example, the fixed cost is 200 PHP.

<p>False (B)</p> Signup and view all the answers

What is the total cost of the function if there are 50 attendees?

<p>22,000</p> Signup and view all the answers

A linear function is defined by the formula f(x) = ______x + ______, where m and b are real numbers.

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What is the maximum number of times the owner can increase the price to maximize his income?

<p>15 (A)</p> Signup and view all the answers

The owner's maximum income is P3,125,000 when the price is increased 15 times.

<p>True (A)</p> Signup and view all the answers

What is the function used to calculate the owner's income (I) based on the number of price increases (x)?

<p>I(x) = -5000x + 15000x + 2000000</p> Signup and view all the answers

What is the first step when graphing a rational function?

<p>Determine the vertical, horizontal, and oblique asymptotes (A)</p> Signup and view all the answers

The function 𝐴(𝑥) = 300𝑥 − 300 calculates the ______ to be paid.

<p>amount</p> Signup and view all the answers

A rational function can have both a horizontal asymptote and an oblique asymptote.

<p>False (B)</p> Signup and view all the answers

Match the function with its corresponding description:

<p>I(x) = -5000x + 15000x + 2000000 = Calculates the owner's income based on the number of price increases A(x) = 300x - 300 = Calculates the amount to be paid based on the number of price increases</p> Signup and view all the answers

What is the value of 𝐴(20) ?

<p>5700 (C)</p> Signup and view all the answers

What is the equation of the oblique asymptote of the rational function f(x) = (x^2 + 2x + 1)/(x - 1)?

<p>y = x + 3</p> Signup and view all the answers

What is the value of 𝑔(−1) in the function 𝑔(𝑥) = − 𝑥 − 𝑥 + 1 ?

<p>-2</p> Signup and view all the answers

A ______ is a vertical line that the graph of a function approaches but never crosses.

<p>vertical asymptote</p> Signup and view all the answers

Match the following terms with their definitions:

<p>Vertical asymptote = A horizontal line that the graph of a function approaches as x approaches positive or negative infinity. Horizontal asymptote = A vertical line that the graph of a function approaches but never crosses. Oblique asymptote = A slanted line that the graph of a function approaches as x approaches positive or negative infinity. Hole = A point in the graph of a function where the function is undefined, but the graph has a gap.</p> Signup and view all the answers

The function 𝑔(𝑥) = − 𝑥 − 𝑥 + 1 always yields positive values for any value of x.

<p>False (B)</p> Signup and view all the answers

How do you find the x-intercepts of a rational function?

<p>Set the numerator equal to zero and solve for x (D)</p> Signup and view all the answers

The inverse of a function always exists.

<p>False (B)</p> Signup and view all the answers

Explain the process of finding the inverse of a function.

<p>To find the inverse of a function, replace f(x) with y, swap x and y, and then solve for y. The resulting equation represents the inverse function, denoted by f^-1(x).</p> Signup and view all the answers

If $log_b m = log_b n$, what can we conclude about the values of m and n?

<p>m = n (B)</p> Signup and view all the answers

The value of $log_b a$ can never be negative.

<p>False (B)</p> Signup and view all the answers

What is the base in the logarithmic expression $log_3 27$?

<p>3</p> Signup and view all the answers

The equation $log_b a = c$ can be rewritten in exponential form as ______ = ______.

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The inverse of a one-to-one function will always also be a one-to-one function.

<p>True (A)</p> Signup and view all the answers

The domain of the inverse function $f^{-1}(x)$ is equal to the ______ of the original function $f(x)$.

<p>range</p> Signup and view all the answers

Which of the following represents the correct step-by-step process for finding the inverse of a function, given the function $f(x)$ ?

<ol> <li>Replace $f(x)$ with $y$.</li> <li>Interchange $x$ and $y$.</li> <li>Solve the equation for $y$.</li> <li>Replace $y$ with $f^{-1}(x)$. (D)</li> </ol> Signup and view all the answers

Given the function $f(x) = 2x - 5$, what is the expression for its inverse function, $f^{-1}(x)$ ?

<p>$f^{-1}(x) = (x + 5) / 2$</p> Signup and view all the answers

Match the following pairs of functions to determine which are inverses of each other:

<p>$f(x) = x^3$ = $g(x) = \sqrt[3]{x}$ $f(x) = 2x + 1$ = $g(x) = rac{x - 1}{2}$ $f(x) = rac{1}{x}$ = $g(x) = x$ $f(x) = 10^x$ = $g(x) = log_{10}(x)$</p> Signup and view all the answers

The graph of a function and its inverse are always symmetrical about the line $y = x$.

<p>True (A)</p> Signup and view all the answers

What is the key principle behind solving exponential equations like $2^x = 8$ ?

<p>Express both sides of the equation as powers of the same base.</p> Signup and view all the answers

The equation $3^{x-2} = 27$ is an example of a(n) ______ equation.

<p>exponential</p> Signup and view all the answers

When solving logarithmic inequalities, what is the first crucial step to ensure?

<p>Check if the arguments of the logarithms are positive (A)</p> Signup and view all the answers

The property of equality for logarithmic equations states that 𝑙𝑜𝑔𝑏𝑚 = 𝑙𝑜𝑔𝑏𝑛 if and only if _____.

<p>m = n</p> Signup and view all the answers

The value of 𝑙𝑜𝑔𝑏𝑎 can never be negative.

<p>False (B)</p> Signup and view all the answers

What is the exponential form of the logarithmic equation 𝑙𝑜𝑔𝑏𝑎 = 𝑐 ?

<p>𝑏𝑐 = 𝑎</p> Signup and view all the answers

Match the following logarithmic properties with their descriptions:

<p>𝑙𝑜𝑔𝑏1 = 0 = The logarithm of 1 to any base is always 0 𝑙𝑜𝑔𝑏𝑏 = 1 = The logarithm of a number to the same base is always 1 𝑙𝑜𝑔𝑏𝑏^𝑥 = 𝑥 = The logarithm of the base raised to a power equals the power 𝑙𝑜𝑔𝑏𝑚 + 𝑙𝑜𝑔𝑏𝑛 = 𝑙𝑜𝑔𝑏(𝑚𝑛) = The logarithm of the product of two numbers is the sum of the logarithms of the numbers 𝑙𝑜𝑔𝑏𝑚 − 𝑙𝑜𝑔𝑏𝑛 = 𝑙𝑜𝑔𝑏(𝑚/𝑛) = The logarithm of the quotient of two numbers is the difference of the logarithms of the numbers</p> Signup and view all the answers

The total cost of a function, denoted by C(x), can be represented by the formula C(x) = mx + b, where m represents the ______ cost per attendee and b represents the ______ cost.

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What is the expression for the overall cost function C as a function of x?

<p>C(x) = 200x + 12,000 (A)</p> Signup and view all the answers

The identity function is defined as f(x) = x.

<p>True (A)</p> Signup and view all the answers

What will be the total cost when there are 50 guests?

<p>22,000 PHP</p> Signup and view all the answers

The slope of a linear function is represented by the letter _____ in the equation f(x) = mx + b.

<p>m</p> Signup and view all the answers

Match the following concepts with their correct definitions:

<p>Linear Function = A function of the form f(x) = mx + b Identity Function = A linear function where f(x) = x Piecewise Function = A function defined by multiple sub-functions Slope = The rate of change in a linear function</p> Signup and view all the answers

What happens to the expected attendance if the ticket price is increased by 20 PHP?

<p>The attendance will decrease by 250. (A)</p> Signup and view all the answers

The greatest possible income received by the owner can be determined by a piecewise function.

<p>True (A)</p> Signup and view all the answers

What is the cost per attendee if the total fixed cost is 12,000 PHP?

<p>200 PHP</p> Signup and view all the answers

Flashcards

Logarithmic Function

A function of the form f(x) = log_b(x), where b is the base.

Logarithm of 2401

log_4(2401) = 4 means 4 raised to the power of 4 equals 2401.

Sound Intensity (Decibels)

In acoustics, measured in decibels (dB) using the formula D = 10 * log(I/I_0).

pH Level

A measure of acidity in a solution, expressed as pH = -log[H+].

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Richter Scale

A logarithmic scale used to measure earthquake magnitude.

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Overall Cost Function

C as a function of x represents total expenses.

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C(x) Calculation

C(x) = 200x + 12,000 for guests.

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Linear Function

A function expressed as f(x) = mx + b.

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Identity Function

A linear function where f(x) = x (m=1, b=0).

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Slope Formula

Slope m calculated as (y2 - y1) / (x2 - x1).

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Piecewise Function

Function defined by different sub-functions for intervals.

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Cost of Event

Calculated by multiplying price per guest by number of guests.

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Sample Problem

Model income changes with ticket price increase and attendance decrease.

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Maximum Price Increase

The highest number of times the owner can raise the price.

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Income Function I(x)

The function calculating the owner's income based on price increase.

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Substitution Method

Replacing a variable in a function with a specific value.

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I(15)

The income calculated when x is substituted with 15.

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Final Income Value

The final amount calculated for I(15) which is P3,125,000.

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Function Evaluation g(x)

Finding the value of the function g for different inputs.

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Function g(-1)

The output of the function g when x is -1.

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Proper Representations

Using correct variables to depict quantities in problems.

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Interchanging x and y

Swapping the values of x and y in a function to find the inverse.

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Finding new y

Solving for the new value of y after interchanging x and y.

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Inverse function definition

A function is an inverse if f(g(x)) = x and g(f(x)) = x for all x in their domains.

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One-to-one function

A function where each output is unique to one input, crucial for an inverse to exist.

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Domain and Range of Inverse

The domain of an inverse function is the range of the original function, and vice versa.

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Graph of the inverse

The graph of an inverse function is the reflection of the original function across the line y = x.

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Exponential equations

Equations that involve variables in the exponent; solved by matching bases.

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Power of the same base

When solving exponential equations, rewrite both sides with a common base.

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Vertical Asymptote

A vertical line where the function approaches infinity, often where the denominator is zero.

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Horizontal Asymptote

A horizontal line that the graph approaches as x tends to infinity.

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Oblique Asymptote

A slanting line that the function approaches, used when the degree of the numerator is one more than the degree of the denominator.

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Zeroes of a Function

The x-values where the function's output is zero, also known as roots.

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Intercepts

Points where the graph intersects the axes; includes x-intercepts and y-intercepts.

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Finding the Inverse

To find the inverse of a function, replace f(x) with y and solve for x in terms of y.

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Graphing Rational Functions

A method to represent functions involving a ratio of polynomials using asymptotes and intercepts.

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Holes in a Function

Points in the graph where the function is undefined, usually from common factors in the numerator and denominator.

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Logarithmic Property of Equality

For log_b(m) = log_b(n), m must equal n for b > 0 and b ≠ 1.

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Logarithmic to Exponential Conversion

log_b(a) = c is equivalent to b^c = a for b > 0 and b ≠ 1.

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Valid Logarithmic Arguments

In log_b(a), a must be positive; that is, a > 0.

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Logarithmic Inequality Basics

For b > 1: if log_b(m) < log_b(n), then m < n.

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Logarithmic Inequality for 0 < b < 1

If 0 < b < 1, log_b(m) < log_b(n) means m > n.

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Logarithm of 1

log_b(1) = 0 for any base b > 0 and b ≠ 1.

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Logarithm of b

log_b(b) = 1 indicates the base log equals one.

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Defined Logarithmic Values

Ensure that logarithmic expressions are defined in any equation.

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Study Notes

Relations

  • A relation is a correspondence between two sets, often represented as ordered pairs.
  • The domain of a relation is the set of first coordinates (inputs).
  • The range of a relation is the set of second coordinates (outputs).
  • Example: {(1, Pepsi), (2, Tropicana), (3, Coke)} shows a relation where 1 corresponds to Pepsi, 2 to Tropicana, and 3 to Coke.

Possible Types of Relations

  • One-to-One: Each element of the first set corresponds to exactly one element in the second set, and vice versa.
  • Many-to-One: Multiple elements in the first set correspond to a single element in the second set.
  • One-to-Many: A single element in the first set corresponds to multiple elements in the second set.

Functions

  • A function is a special type of relation where each element in the domain (input) corresponds to exactly one element in the range (output).
  • Only one-to-one and many-to-one relations are considered functions. One-to-many relations are not functions.

Vertical Line Test

  • A graph represents a function if and only if no vertical line intersects the graph at more than one point.
  • If a vertical line intersects the graph in more than one place, the graph does not represent a function.

Types of Functions

  • Linear: f(x) = mx + b
  • Quadratic: f(x) = ax² + bx + c
  • Cubic: f(x) = ax³ + bx² + cx + d
  • Piecewise: A function defined by multiple sub-functions, applying to intervals of the domain.
  • Rational: f(x) = n(x)/d(x), where n(x) and d(x) are polynomials, and d(x) ≠ 0
  • Exponential: f(x) = bˣ, where b > 0 and b ≠ 1
  • Logarithmic: f(x) = logₐx, where a > 0 and a ≠ 1

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