College Algebra Unit 2

Equation of a circle

(x-h)²+(y-k)²=r² (h, k) is the center, r is the radius

Steps to find the inverse of a function

1) Replace f(x) with y in the equation for f(x) 2) Change the x's to y's and the y to an x 3) Solve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If it does define y as a function of x, the function f has an inverse function 4) Change y to f^ -1(x). If f has an inverse function, replace y in step 3 with f-1(x). Verify the result by showing f(f-1 (x)) = x and f-1(f (x)) = x

How to input a matrix into the calculator

1. Click 2nd and x-1 to go to matrix 2. Go to EDIT 3. Hit Enter 4. Type the numbers of rows and columns into the top 5. Type the numbers of the equations (should be x+y=# style) 6. Click 2nd and MODE to quit 7. Click 2nd and x-1 to go to matrix 8. Go to MATH 9. Scroll down until you find rref( 10. Click 2nd and x-1 again 11. Hit Enter

Solving linear systems with the addition method (also called the elimination method)

1. If necessary, rewrite both equations in the form ax+by=c 2. If necessary, multiply either equation or both equations by appropriate nonzero numbers so the sum of the x-coefficients or the sum of the y-coefficients is 0 3. Add the equations in step 2. The sum is an equation in one variable 4. Solve the equation in one variable 5. Back-substitute the value obtained in step 4 into either of the given original equations and solve for the other variable 6. Check the solution in both of the original equations

How to solve linear equations with substitution

1. Solve either of the equations for one variable in terms of the other 2. Substitute the expression found in Step 1 into the other equation. This will result in an equation with one variable 3. Solve the equation with one variable 4. Back-substitute the value found in Step 3 into one of the original equations. (You can use the equation found in Step 1). Simplify and find the value of the remaining variable 5. Check the proposed solution in both of the system''s given equations

The domain of f(g(x)) is the set of all x such that

1. x is in the domain of g and 2. g(x) is in the domain of f

Consistent system definition

A system with a least one solution. This can be lines that intersect once (meaning they have one solution), or the same line (meaning they have infinitely many solutions)

Inconsistent system definition

A system with no solution. The graphs are parallel. If you try to solve an inconsistent solution with addition or subtraction, you will eliminate both variables. A false statement will result

Linear systems

All equations in the form of ax+by=c are straight lines when graphed. Two equations are called a system of linear equations or a linear system. A solution to a system of linear equations in two variables is an ordered pair that satisfies both equations in the system

Dependent system definition

An equation with infinitely many solutions. If you try to solve a dependent system with addition or subtraction, you will eliminate both variables and a true statement will result

Graphs of f and f-1

Because inverse functions have ordered pairs with the coordinates interchanged, if point (a,b) is on the graph of f, the point (b,a) is on the graph of f-1. The points (a,b) and (b,a) are symmetric with respect to the line y=x. Thus, the graph of f-1 is a reflection of the graph of f about the line y=x

Inverse functions definition

Functions that undo each other are inverse functions So let f and g be two functions such that f(g(x)) = x for every x in the domain of g and g(f(x)) = x for every x in the domain of f

When substituting coordinates of a test point into an inequality....

If a true statement results, shade the half-plane containing the test point. If a false statements results, shade the half-plane not containing the test point

No solutions in a matrix in a calculator

If it has a repeating number and an arrow beside in the matrix, the system has no solutions, or ∅. The answer is ∅ Example: [1 -0.3333333333 > [0 0

Infinite solutions in a matrix in a calculator

If the bottom row is all 0 but the top row has other numbers in it, the system has infinite solutions. The answer includes the first equation with (x,y) in front of it Example: [1 -2 4] [0 0 0]

Domain of a function

If the function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of f(x) is a real number. The domain of functions are usually real numbers or (∞,-∞).

Restrictions on domain

If you have a fraction with a variable in the denominator, you must find the values that make the denominator equal to 0 and exclude those values from the domain. If you have an even root function (like a square root) with the variable inside the radical, you must set the radicand (the part inside the radical) greater or equal to 0 and solve to find the domain. Answers can be written in set builder notation or in interval notation.

Inverse functions rules

Not all functions have an inverse function. To have an inverse function, a function must be one-to-one. The Vertical Line Test is used to to determine if a function has an inverse. If a horizontal line intersects the graph of a function at most once, the function is one-to-one and has an inverse

A system of linear equations can have

One solution, no solutions, or infinite solutions

Combining functions

One way to combine functions is to form the composition of one with another. This composition is denoted as f · g and is read as ''f composed as g'' or ''f of g.'' It is putting one function into another function (not multiplication). The domain of a composition must first come from the domain of the inner function. For example, if you have (f · g)(x) you should rewrite it as f(g(x)). The inner function is g(x). So first find the domain of the inner function g(x) to find any restrictions and then solve f(g(x)) to find any further restrictions. The domain must satisfy all restrictions.

How to show inverse functions

Put one equation into the other. Simplify or solve the equation (if possible). If they are inverses, they will simplify down to just x. If they do not, they are not inverses of each other

When multiplying or dividing by a negative number around an inequality sign

Remember to switch the direction of the inequality sign

Rows and columns in matrices

Rows are the horizontal lines in matrices Columns are the vertical lines in matrices

If the line is horizontal (y = #)

Shade above for > and below for <

If y > mx+b

Shade the area above the line y > mx+b

If y < mx+b

Shade the area below the line y < mx+b

If the line is vertical (x = #)

Shade to the right for > and to the left for <

G and F function formulas

Sum: (f+g)(x) = f(x)+g(x) Difference: (f-g)(x) = f(x)-g(x) Product: (fg)(x) = f(x)*g(x) Quotient: (f/g)(x) = f(x)/g(x), where g(x) ≠ 0

How to read answers in a matrix

The answers are the very last numbers in the set. The top one is x, the second one is y, and the third one is z Example: in the answer, x = 4, y = 3, z = 4

Inverse functions notes

The inverse of addition is subtraction. The inverse of multiplication is division. The inverse of squaring a value is taking the square root. Function g is the inverse of function f and is denoted by inverse function f-1 (read f-inverse). Thus, f(f-1 (x)) = x and f-1(f-1 (x)) = x. The domain of f is equal to the range of f-1, and the range of f is equal to the domain of f-1

If the inequality contains a < or >

The line is made of a dashed line

If the inequality contains a ≥ or ≤

The line is made of a solid line

Break-even point definition

The point of intersections of the graphs of the revenue and cost functions is called the break-even point. The x-coordinate represents the number of units a company must produce and sell so the money coming in, the revenue, is equal to the money going out, the cost. The y-coordinate of the break-even point gives the amount of money coming in and going out

Profit function

The profit, P(x), generated after producing and selling x units of a product is given by the profit function P(x) = R(x)-C(x) Where R and C are the revenue and cost functions, respectively

Graphing systems of linear inequalities

The solution set of a system of linear inequalities in two variables is the set of all ordered pairs that satisfy each inequality in the system. Thus, to graph a system of inequalities in two variables, begin by graphing each individual inequality in the same rectangular coordinate system. Then find the region, if there is one, that is common to every graph in the system. This region of intersection gives a picture of the system´s solution set

Functions from verbal descriptions

There is no rigid step-by-step procudeure that can be used to construct a function from verbal descriptions. Read the problem carefully. Attempt to write a critical sentence that describes the function's conditions in terms of its independent variable, x

Function with no inverse

This function fails the horizontal line test, so it has no inverse

Inverse function

This function passes the horizontal line test, so it has an inverse

Decomposing functions

When you write a composite function, you ''compose'' two functions to create a new function. You can also reverse the process, ''decomposing'' a given function and expressing it as a composition of two functions. Consider the function h defined by h(x) = (3x²-4x+1)⁵/g If you think of 3x²-4x+1 as the inner part of the function, you can tell it´s all being raised to the 5th power, which is the outer part of the function So g(x) = 3x²-4x+1 and f(x) = x⁵. If you find f(g(x)), you have h(x) = (3x²-4x+1)⁵

Distance formula

d = rt (distance = rate * time)

If an object moves with average velocity (v), the distance (s) covered in time (t) is given by the formula

s= vt Objects that move in accordance with this formula are said to be in uniform motion

Meaning of ∅

∅ means there are no points in the system that satisfy all the inequalities in the system. If there is no overlapping region, we say the solution set is ∅, the empty set