Algebra 1: Functions and Matrix's

Function

A rule that establishes a relationship between two quantities, called the input and the output. For each input, there is exactly one output.

Domain

All x values (input) and independent variable

How to solve function notation as f(#)

Solving function notation as f(#): 1) look at equation given and the part that shows what you need to find, which in this case will always be the y. 2) plug in number representing x found in the equation. 3) solve. 4) write the answer in function notation form. EX: k(x)=-3x² +x; find k(-1). k(x) becomes k(-1) as it told us what our x-value is, and it's also saying we solve for y. fill in -1 for every x. You don't have too but it helps when solving it. k(x)=-3x² -1. k(-1)=-3(-1)²-1. Pemdas, do exponents first. -1 to the second power is 1. k(-1)=-3(1)-1. then do parentheses. -3 times 1 is -3. k(-1) = -3-1. Answer: k(-1) =-4.

piecewise

a function made up from pieces of other functions

matrix

a matrix is a rectangular array of variables or constants in horizontal rows and vertical columns, usually enclosed in brackets. Each value in a matrix is called an element. The dimensions of a matrix is the size of the matrix measured in rows and columns. EX: a 3 by 2 matrix is 3 rows and 2 columns. Always identify the dimensions of a matrix!

function notation

a way to name a function that is defined by an equation. EX: f(x) = mx+b. f(x) = y, its the same thing! Note that the 'f' and 'x' can be replaced with any other letter and it will still equal y.

Range

all y values (output) and dependent variable

calculator hints

ch: when you are typing a number that has an exponet on the calculator, surround that number and its sign with a ( ). It gives you the correct answer! unlike when you type it up for regular math problems, it won't read well on calculator.

how to graph linear functions

graphing linear functions: 1) find the domain and range of the function that is shown in a graph. 2) find f(x)=# according to what the problem says. 3) find f(#) according to what the problem says. 4) solve any other question asked with the information you now have, depending on the problem. EX: you have a graph about a bike trip which started at 9 am and ended at 1 pm. the domain is 0 ≤ x ≤ 4 because the points on the x are from 0 to 4. the range is 0 ≤ y ≤ 20 because the y-axis is covered from 0 to 20. Find f(2), this is telling us that x=2 and y=?. in order to figure this out we use the graph, locate where 2 is and see where 2 passes the y-axis which is 20. answer: f(2)=20. Find f(x)=20, this is telling us that y=20 and x=? We do the opposite in finding f(2). So we go to where 20 is on the y-axis and locate where it intersects the x-axis which is 2. The answer is the same, f(2)=20. Hint: there can always be more than one answer to finding the x, never the y!

infinity d and r

infinity domain and range: arrows that don't have a point in front of them can go to infinity on the graph. the domain is part of the x-axis so that infinity would be positive based on where the point is located most of the time. the range is the y-axis which is vertical and it can be both positive and negative infinity. Negative infinity is first and positive infinity is second because we start from the bottom part of the y-axis, which is where negative numbers are. Then we make our way to the top where positive numbers come into place, representing positive infinity.

how to solve addition matrix

solving addition matrix: 1) You can only add matrices of the same order, which is like the same size. EX: one matrix is 3 by 2, the other is also 3 by 2. That can be solved. If a matrix is 3 by 2, and the other is 2 by 3, that is undefined. 2) add the corresponding elements by row. It is easier to finish one row than move on to the next one afterward to solve. EX: 3+4=7. 4+1=5. the first element in the first matrix is always added with the first element of the second matrix. 8+0=8. 6 + -9 = -3. The second element in the row of the first matrix is always added with the second element of the second matrix. 3)write your answer in the correct size of the two elements. Make a space in between each element as you write them.

how to solve function notation as f(x) = #

solving function notation as f(x) = #: 1) look at the equation given. 2) find the x value by subbing in what # f(x) ='s. 3) rewrite answer in function notation form. EX: f(x) = 3x+1. This is telling us that our equation is y=3x+1. f(x)=10. This is telling us that y=10 but x=?. We must plug 10 in for y and solve. 10=3x+1. Minus 1 on both sides. 9=3x. divide by 3. 3=x. Answer: f(3)=10. This is saying that when x is 3, y is 10 for this function.

how to solve functions through mapping diagram

solving functions through mapping diagram: 1) label the input and output sections. 2) make sure that for each input there is exactly one output. You CANT have repeated x-values(inputs)! You CAN have repeated y-values (outputs). One input CANT have more than one output! 3)write "its a function" or "not a function" and explain why.

How to solve functions through table

solving functions through table: 1) slope = change in y/change in x. Check if there is a constant rate of change. EX: slope = 4/2. slope = 2. HINT: if the numbers don't seem constant but for each section its the same number, then its just 1. 2) A point. locate a point with a zero in it or use the point that has smaller numbers to use in the next step. 3) make a rule(equation) for the function. You use y=mx+b to do so. EX: slope = 2. I choose point (3,6). 6=2(3)+b. 6=6+b. minus 6 on both sides. answer: y=2x+0.

how to solve matrix scalar with variables

solving matrix scalar with variables: 1) you have a variable in front of the matrix. You either look at the matrix(s) and see what the answer is, based on the solution. Or you look at one row and its solution and divide. EX: k is in front of the matrix. The first matrix has 2 in the first row. the solution is 6. 2k=6. divide by 2. k=3. 2) you also might have to find the value of both x or y. You would multiply the number in front of that matrix only. Don't multiply the scalar for each one since the scalar is not attached to each matrix! Once you finished multiplying the scalar number to its specific matrix, you would take the numbers that have a variable and follow what their specific row says. Finally, you would do the order of operations to find x and y. EX: 2 is the number in front of the matrix. Times to each matrix. {2x 4} - {6 y} = {5 3}. 2x-6=5. plus 6 on both sides. 2x=11. x=5.5. 4-y=3. minus 4 on both sides. -y=-1. divide by -1. y=1.

how to solve multiply matrix

solving multiply matrix: 1) identify the dimensions of the two matrix(s). If the two numbers in the middle are the same, then you can multiply. This is used for all multiplication matrix properties. EX: 2 by 2 and 2 by 2. the matching twos in the center make it possible for the matrix to be solved. the matrix size for the answer is 2 by 2, using the numbers opposite to the numbers in the middle. these numbers CAN be the same. EX: if its 2 by 3 and 2 by 3, then it can't work. 2) multiply the row to a single column and so on. You always multiply the row to the column! EX {1 2 3 4 5 6} x {7 8 9 10 11 12}. 1, 2, 3 are a row. 4, 5, and 6 are a row. you times 1, 2 and 3 to 7, 9, and 11. 1 x 7, 2 x9, and so on. 3) add the multiplication results together. EX: 1x7=7.2x9=18.3x11=33. 7+18+33=56. My answer is 56. 4) you do the same thing, multiply 1, 2, and 3 but with the second column 8, 10, and 12, and add the multiplication results together to get another answer. 5) you do the same steps in 2, 3, and 4 but for the second row, using the same columns. Use a calculator if needed or write the steps out, that is ok!

how to solve multiply scalar

solving multiply scalar. scalar is a matrix with a number in front of it. steps: 1) use the distributive property. we multiply each element of the matrix by the number in front. EX: -2 is scalar. the numbers in the matrix are -6, -3, -1, and -2. answer: { 12 6 2 4}. Hint: when multiplying a # when there's a variable in front of the element, just put the scalar number next to that variable.

How to solve piecewise functions

solving piecewise functions: 1) find f(#) using the rules assigned to it. these rules will use inequalities to represent them, followed by the equation. You can label them as "a" or "b" to separate which is which. 2) whatever the inequality is showing, you must put that number first into the table you make, despite it being accurate or not to what the inequality sign is showing. You represent such points with an open circle. the rest that fits the criteria of the inequality sign, they get a closed circle. We just found our x-values. 3) sub in x-values into the inequality problem. That's how you find the y. afterward, plot them on the graph. 4) determine if your line stops at its point or needs an arrow, which is found using the inequality sign. Or if a point prevents it from continuing. 5) determine if the function is continuous or not by seeing if the lines intersect at one point or if the line can be drawn with one stroke only. If not, then it's not continuous. Hint: if the numbers you have for the outputs(y) are mainly negatives than your arrow must be towards the negative side of the graph despite the inequality sign. EX: (following the steps above here). 0 > 0. I solve for zero three times using 0, 1, and 2, as my x's. 0 gets an open circle while 1 and 2 have closed circles. The y's for each one is (0,-3), (1,1), and (2,-1). Since there's a negative, the arrow won't go upward but instead downward.

how to solve subtraction matrix

solving subtraction matrix: 1) You can only subtract matrices of the same same size (# by #). if two matrices are not the same size, then it's undefined. 2) subtract the corresponding elements by row. 3) if one number is negative then you turn it into a plus sign and continue solving that element. EX: 7--3 turns into 7+3 = 10. 3-0=3. 5-8=-3. and so on. The first element in the first matrix is always subtracted with the first element of the second matrix. The second element in the row of the first matrix is always subtracted with the second element of the second matrix. 4) write your answer in the correct size of the two elements. Make a space in between each element as you write them.