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Comprehensive General Mathematics Notes

sam ⋆𐙚₊˚⊹♡

@samantha.badiola_

Ever wonder how math connects to real life? Functions and... Ipakita pa

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General Mathematics
Lesson 1.1: Functions &
Relations
DEFINITION OF A RELATION:
- A relation is a rule that relates
values from a set of val

Understanding Relations and Functions

Think of a relation as a matchmaker that connects values from one group (the domain) to another group (the range). It's basically a set of ordered pairs like (x,y) where x gets paired with y.

Now here's where it gets interesting - a function is a special type of relation with one important rule: each x-value can only be matched with one y-value. No cheating allowed! If you see an x-value trying to connect to multiple y-values, it's not a function.

For example, f = {(1,2), (2,3), (3,5)} is a function because each x has its own unique y. But g = {(1,3), (1,4), (2,5)} isn't a function because x = 1 is trying to pair with both 3 and 4.

Quick Tip: Functions use the notation f(x) = y, which reads as "f of x equals y." This just means when you input x, you get y as the output.

Spotting Functions with Visual Tests

Mapping diagrams show functions as arrows connecting domain values to range values. If any domain value has multiple arrows pointing out, it's not a function - remember, one input can only have one output!

The vertical line test is your best friend for checking graphs. Draw imaginary vertical lines across the graph - if any line touches the curve more than once, you don't have a function. It's that simple!

Tables of values organize functions neatly, showing x and y pairs in columns. These make it super easy to see the relationship between inputs and outputs at a glance.

Piecewise functions are like having different rules for different situations. They use multiple equations depending on which domain you're working in - think of them as "if-then" statements in math form.

Remember: Functions are everywhere in real life - from calculating grades based on test scores to determining taxi fares based on distance traveled!

Evaluating Functions Like a Pro

Evaluating functions means finding the output when you plug in a specific input value. It's like following a recipe - substitute the given number for x and calculate the result.

For regular functions like f(x) = x² - 4x + 4, just replace every x with your given number. If you need f(2), substitute 2 for x: f(2) = 2² - 4(2) + 4 = 0. Easy math!

Piecewise functions require an extra step - first figure out which equation to use based on your x-value, then substitute and solve. Check the conditions $like x > 5 or x = 5$ to pick the right formula.

Practice makes perfect with these problems. Start with simple substitutions, then work your way up to more complex piecewise functions as you build confidence.

Pro Strategy: Always double-check which piece of a piecewise function applies to your x-value before calculating - this prevents the most common mistakes!

Function Operations Made Simple

Just like regular numbers, functions can be added, subtracted, multiplied, and divided. The key is performing the operation on the function expressions themselves.

For addition and subtraction, combine like terms: $f+g$ (x) = f(x) + g(x). If f(x) = x+1 and g(x) = x²+2x-8, then $f+g$ (x) = $x+1$ + $x²+2x-8$ = x² + 3x - 7.

Multiplication works the same way: (f×g)(x) = f(x) × g(x). Just multiply the entire expressions together and simplify. Division follows suit, but watch out for values that make the denominator zero!

Composite functions like (f∘g)(x) = f(g(x)) mean you're plugging one function into another. Work from the inside out - first find g(x), then use that result as the input for f(x).

Memory Hack: Think of composite functions like nesting dolls - you work your way from the innermost function outward!

Mastering Composite Functions

Composite functions might look intimidating, but they're just functions within functions. The notation (g∘f)(x) means "g of f of x" - you calculate f(x) first, then plug that answer into g(x).

Let's say f(x) = x + 3 and g(x) = 2x - 7. To find (g∘f)(x), substitute the entire f(x) expression into g(x): g(f(x)) = g $x + 3$ = 2 $x + 3$ - 7 = 2x + 6 - 7 = 2x - 1.

Notice that (g∘f)(x) usually gives a different result than (f∘g)(x) - order matters! Always work carefully through each step and double-check your algebra.

These composite functions show up everywhere in real applications - like converting temperature from Celsius to Fahrenheit, then adjusting for altitude effects.

Success Tip: When dealing with composite functions, take your time with substitution and use parentheses to avoid sign errors!

Functions as Real-World Models

Functions aren't just abstract math - they're powerful tools for modeling real-world relationships! Mathematical modeling helps us understand how different variables affect each other in everyday situations.

Direct variation $y = kx$ shows relationships where both variables increase or decrease together - like distance and time when driving at constant speed. Inverse variation $y = k/x$ describes opposite relationships - as one increases, the other decreases, like speed and travel time.

Joint variation $z = kxy$ involves three variables where z depends on both x and y together - think about how the area of a rectangle depends on both length and width.

The constant of variation (k) is crucial - it tells you exactly how strong the relationship is. Once you find k using given values, you can predict what happens in new situations using your mathematical model.

Real-Life Connection: These variations explain everything from how your electricity bill relates to usage, to how gravitational force changes with distance between objects!

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Na-update Mar 19, 2026

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6 mga pahina

Comprehensive General Mathematics Notes

sam ⋆𐙚₊˚⊹♡

@samantha.badiola_

Ever wonder how math connects to real life? Functions and relations are everywhere - from calculating your phone bill to predicting population growth. These mathematical tools help us understand patterns and solve everyday problems in a systematic way.

Mag-sign up para makita ang contentLibre ito!

Access sa lahat ng dokumento

Pagbutihin ang iyong mga grado

Sumali sa milyong mga estudyante

Understanding Relations and Functions

For example, f = {(1,2), (2,3), (3,5)} is a function because each x has its own unique y. But g = {(1,3), (1,4), (2,5)} isn't a function because x = 1 is trying to pair with both 3 and 4.

Quick Tip: Functions use the notation f(x) = y, which reads as "f of x equals y." This just means when you input x, you get y as the output.

Mag-sign up para makita ang contentLibre ito!

Access sa lahat ng dokumento

Pagbutihin ang iyong mga grado

Sumali sa milyong mga estudyante

Spotting Functions with Visual Tests

Tables of values organize functions neatly, showing x and y pairs in columns. These make it super easy to see the relationship between inputs and outputs at a glance.

Remember: Functions are everywhere in real life - from calculating grades based on test scores to determining taxi fares based on distance traveled!

Mag-sign up para makita ang contentLibre ito!

Access sa lahat ng dokumento

Pagbutihin ang iyong mga grado

Sumali sa milyong mga estudyante

Evaluating Functions Like a Pro

Evaluating functions means finding the output when you plug in a specific input value. It's like following a recipe - substitute the given number for x and calculate the result.

For regular functions like f(x) = x² - 4x + 4, just replace every x with your given number. If you need f(2), substitute 2 for x: f(2) = 2² - 4(2) + 4 = 0. Easy math!

Practice makes perfect with these problems. Start with simple substitutions, then work your way up to more complex piecewise functions as you build confidence.

Pro Strategy: Always double-check which piece of a piecewise function applies to your x-value before calculating - this prevents the most common mistakes!

Mag-sign up para makita ang contentLibre ito!

Access sa lahat ng dokumento

Pagbutihin ang iyong mga grado

Sumali sa milyong mga estudyante

Function Operations Made Simple

Just like regular numbers, functions can be added, subtracted, multiplied, and divided. The key is performing the operation on the function expressions themselves.

For addition and subtraction, combine like terms: $f+g$ (x) = f(x) + g(x). If f(x) = x+1 and g(x) = x²+2x-8, then $f+g$ (x) = $x+1$ + $x²+2x-8$ = x² + 3x - 7.

Composite functions like (f∘g)(x) = f(g(x)) mean you're plugging one function into another. Work from the inside out - first find g(x), then use that result as the input for f(x).

Memory Hack: Think of composite functions like nesting dolls - you work your way from the innermost function outward!

Mag-sign up para makita ang contentLibre ito!

Access sa lahat ng dokumento

Pagbutihin ang iyong mga grado

Sumali sa milyong mga estudyante

Mastering Composite Functions

Composite functions might look intimidating, but they're just functions within functions. The notation (g∘f)(x) means "g of f of x" - you calculate f(x) first, then plug that answer into g(x).

Let's say f(x) = x + 3 and g(x) = 2x - 7. To find (g∘f)(x), substitute the entire f(x) expression into g(x): g(f(x)) = g $x + 3$ = 2 $x + 3$ - 7 = 2x + 6 - 7 = 2x - 1.

Notice that (g∘f)(x) usually gives a different result than (f∘g)(x) - order matters! Always work carefully through each step and double-check your algebra.

These composite functions show up everywhere in real applications - like converting temperature from Celsius to Fahrenheit, then adjusting for altitude effects.

Success Tip: When dealing with composite functions, take your time with substitution and use parentheses to avoid sign errors!

Mag-sign up para makita ang contentLibre ito!

Access sa lahat ng dokumento

Pagbutihin ang iyong mga grado

Sumali sa milyong mga estudyante

Functions as Real-World Models

Joint variation $z = kxy$ involves three variables where z depends on both x and y together - think about how the area of a rectangle depends on both length and width.

Real-Life Connection: These variations explain everything from how your electricity bill relates to usage, to how gravitational force changes with distance between objects!

Akala namin hindi mo na itatanong...

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Saan ko mada-download ang Knowunity app?

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Talaga bang libre ang Knowunity?

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Stefan S

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Samantha Klich

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Pinakamagandang app sa mundo! walang masabi dahil sobrang ganda nito

Thomas R

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David K

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Sudenaz Ocak

Android user

Sa paaralan, napakahina ko sa math pero salamat sa app, mas mahusay na ako ngayon. Lubos akong nagpapasalamat na ginawa niyo ang app na ito.

Greenlight Bonnie

Android user

Rohan U

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Xander S

iOS user

Elisha

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Paul T

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Stefan S

gumagamit ng iOS

Samantha Klich

Android user

Anna

iOS user

Pinakamagandang app sa mundo! walang masabi dahil sobrang ganda nito

Thomas R

iOS user

Basil

Android user

David K

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Sudenaz Ocak

Android user

Sa paaralan, napakahina ko sa math pero salamat sa app, mas mahusay na ako ngayon. Lubos akong nagpapasalamat na ginawa niyo ang app na ito.

Greenlight Bonnie

Android user

Rohan U

Android user

Xander S

iOS user

Elisha

iOS user

Paul T

iOS user