This week, we learned about function notation, mapping notation, and relations. I’m focusing on function notation because it’s important for understanding other math topics. Here are the steps to follow, with two simple examples:

Steps for Function Notation:

1. Know the function name:
   The name is usually a letter like f, g, or h.
2. Find the input value (x):
   The input is the number you will substitute into the function.
3. Plug in the input:
   Replace x with the given number in the equation.
4. Solve:
   Do the math to find the answer.

Example 1:

Let g(x)=4x−7
Find g(3):

Step 1: The function name is g.
Step 2: The input is x=3
Step 3: Replace x with 3:

g(3)=4(3)−7

Step 4: Solve:

g(3)=12−7=5

Answer: g(3)=5

Example 2:

Let h(x)=x2+2x−1
Find h(−2):

Step 1: The function name is h
Step 2: The input is x=−2
Step 3: Replace x with -2:

h(−2)=(−2)2+2(−2)−1

Step 4: Solve:

h(−2)=4−4−1=−1

Answer: h(−2)=−1

The domain and range in mathematics describe the inputs and outputs of a function:

1. Domain:

The domain is the set of all possible inputs ($x$-values) for which the function is defined. It excludes any values that cause division by zero, negative square roots (in real numbers), or any other undefined operations.

2. Range:

The range is the set of all possible outputs ($y$-values) that the function can produce based on its domain.

Example 1: f(x)=x^2−4

Domain: You can square any real number, so:

Domain=(−∞,∞)

Range: The smallest value of x^2 is 0 (when x=0), so the smallest output of f(x) is:

Range=[−4,∞)

Example 2: f(x)=x+3

Domain: There are no restrictions on $x$, so:

Domain=(−∞,∞)

Range: Adding 3 shifts all the outputs of $x$. Since $x$ can be any real number, the range is also:

Range=(−∞,∞)

Example 1: Finding the x- and y-intercepts for y=2x−6y

Finding the x-intercept:

1. To find the x-intercept, I set y=0 because the x-intercept is where the line crosses the x-axis (where $y$ is zero).
2. Substituting y=0 gives us the equation: 0=2x−6
3. Next, I solved for x by isolating it on one side of the equation: 2x=6 and x=3
4. So, the x-intercept is (3,0)

Finding the y-intercept:

1. To find the y-intercept, I set x=0 because the y-intercept is where the line crosses the y-axis (where x is zero).
2. Substituting x=0 gives us the equation: y=2(0)−6y 
3. So, the y-intercept is (0,−6)

For this equation, I learned that finding intercepts helps us sketch the line on a graph quickly since the intercepts provide two points through which the line passes.

Example 2: Finding the x- and y-intercepts for y=x^2−4

Finding the x-intercepts:

1. I set y=0 to find where the graph crosses the x-axis.
2. This gives us: 0=x2−4
3. To isolate $x$, I added 4 to both sides: x^2=4
4. Then, I took the square root of both sides, giving us two possible x-values: x=2 and x=−2
5. So, the x-intercepts are (2,0) and (−2,0).

Finding the y-intercept:

1. I set x=0 because the y-intercept occurs where $x$ is zero.
2. Substituting x=0 into the equation: y=(0)2−4
3. The y-intercept is (0,−4).

Prime Factorization is the process of breaking down a number into a product of prime numbers (numbers that have no factors other than 1 and themselves). A helpful way to find these prime factors is by using factor trees.

Steps for Prime Factorization

1. Start with the Number: Choose a number you want to factor.
2. Choose Factors: Think of two numbers that multiply to get that number. Write the number at the top, then draw two branches for the factors.
3. Repeat Until All Factors are Prime: For each composite (non-prime) factor, keep breaking it down into smaller factors. Stop when all branches lead to prime numbers.
4. List the Prime Factors: Once all factors are prime, list them.
5. Combine Like Factors with Exponents: Group any repeating prime factors using exponents to simplify the answer.

Example 1: Prime Factorization of 180

1. Start: Write 180.
2. First Split: Choose factors like $60$ and $3$.
3. Continue Factoring:
   • $3$ is already prime.
   • Break $60$ down into $30$ and $2$.
   • $2$ is prime.
   • Break 30 down into 1$5$ and $2$.
   • Break 15 into $5$ and $3$, which are both prime.
4. List Prime Factors: $2$, $2$, 3, $3$, and $5$.
5. Combine with Exponents: 180=2^2⋅3^2⋅5.

Example 2: Prime Factorization of 210

1. Start: Write $210$.
2. First Split: Choose factors like $2$ and 105.
3. Continue Factoring:
   • $2$ is prime.
   • Break 105 into $5$ and 21.
   • 5 is prime.
   • Break 21 into $3$ and $7$, both prime.
4. List Prime Factors: 2, 3, 5, and $7$.
5. Final Answer: 210=2⋅3⋅5⋅7

In class this week, we covered factoring polynomial. A polynomial can be factored by removing or dividing out the greatest common factor (GCF) from each term. The examples below show how we applied this method to different types of problems.

Steps
1. Combine like terms / simplify
2. Factor the resulting polynomial

in my First Ex, I combined like terms and then factored the result. This process involved simplifying the polynomial and finding the common factor.

in my Second Ex, I didn’t need to combine like terms because the expression was already simplified. I directly factored it by identifying the GCF.

Example 1:
5x^2−2x+7=2x^2+8x−7

3x^2+6x=3(x+2x)

Example 2:

9x^4 – 6x^3 + 3x^2

= 3x^2(3x^2 – 2x + 1)

This approach helps simplify complex expressions by breaking them down into more manageable factors.