Key Equations
What is the formula for the inverse of a matrix?
A^{-1}=\frac{1}{\text{det}A}\text{adj}A
Evaluate the following expression
(-8-3)(-3)--2-8(0)-2
33
Come up to the board and draw a graph which is not a function. Why is it not a function?
The graph drawn fails the vertical line test.
Find
A\cup B
A\capB
A={1,2,3}
B={2,c}
A\cup B={1,2,3,c}
A\cap B={2}
What is the formula for an ellipse? What variable is the x-radius? What variable is the y-radius?
\frac{x^2}{a}+\frac{y^2}{b}=1
Find
f\circ g
g\circ f
f(x)=x^2
g(x)=x-4
f\circ g=x^2-8x+16
g\circ f =x^2-4
Come up to the board and draw a graph which has no real solutions. Why does the graph have no real solutions?
The graph drawn should not cross the x-axis.
Come up with an example of an injective map and sketch it on the board.
Needs two sets and a defined relation from one to the other...
What is the formula for an hyperbola?
\frac{x^2}{a}-\frac{y^2}{b}=1
Factor the following
x^3-2x^2+5x-10
(x^2+5)(x-2)
What does the following expression always equal
A A^{-1}
\begin{bmatrix} 1 & 0 \\ 0& 1 \\ \end{bmatrix}
Using only addition, add eight 8s to get the number 1,000
888+88+8+8+8=1000
Evaluate the following
\sqrt{-25}
\sqrt{-75}
5i
5i\sqrt{3}
What are domains of the following functions? Do they have any restrictions, if so, what are they?
f(x)=x+2
g(x)=\frac{1}{x+2}
h(x)=\sqrt{x+2}
1. No restrictions, all real numbers is the domain.
2. One restriction that the denominator cannot equal zero, Domain cannot include -2
3. One restriction that the square root cannot be negative. Domain cannot include any number less than -2.
What does the fundamental theorem of algebra tell us?
It tells us how many solutions a polynomial has by looking at its degree.
Solve the following inequality. Make sure your solution is in interval form!
|-3x+6|\leq 18
(-infty,8]\cup[-4,\infty)
Factor the following
x^2-36
(x-6)(x+6)
Does the following matrix have an inverse? If so, what is it? *Show work.
A=\begin{bmatrix} 7 & 2 \\ 17 & 5 \\ \end{bmatrix}
A=\begin{bmatrix} 5 & -2 \\ -17 & 7 \\ \end{bmatrix}
Is
A\subseteq B
A={x\in\mathbb{N}: x=4\mathbb{N}}
B={x\in\mathbb{N}: x=2\mathbb{N}}
Yes because the cardinality of A is less than B and everything contained in A is contained in B.
The zeros of a polynomial are. -1, 2, and 3. Find the expression for the polynomial. What is its degree. What theorem allowed you to write the expression before expanding.
The linear factorization theorem.
x^3-4x^2+x+6