College Algebra - Putting It All Together

Standard form ax + by = c

Any line can be written in this form. 3x + 5y = 15 x = -4 Vertical line y = 7 Horizontal line

Numerical representation of a function

Table of values. A partial numerical representation of f(x) = 3x is shown. x 0 1 2 3 f(x) 0 3 6 9

Composition of two functions

(gof)(x) = g(f(x)) f(x) = x^3, g(x) = x^2 - 2x + 1 (gof)(2) = g(f(2)) = g(8) = 64 - 16 + 1 = 49 (gof)(x) = g(f(x)) = g(x^3) = (x^3)^2) - 2 (x^3) + 1 = x^6 - 2x^3 + 1

Horizontal line test

If every horizontal line intersects the graph of f at most once, then f is one-to-one.

Intergers

Include the natural numbers, their opposites, and 0 ..., -2, -1, 0, 1, 2,...

Sum of two functions

(f + g)(x) = f(x) + g(x) f(x) = x^2, g(x) = 2x + 1 (f + g)(3) = f(3) + g(3) = 9 + 7 = 16 (f + g)(x) = f(x) + g(x) = x^2 + 2x + 1

Difference of two functions

(f - g)(x) = f(x) - g(x) f(x) = 3x, g(x) = 2x + 1 (f - g)(1) = f(1) - g(1) = 3 - 3 = 0 (f - g)(x) = f(x) - g(x) = 3x - (2x + 1) = x - 1

Quotient of two functions

(f / g)(x) = f(x) / g(x), g(x) not = 0 f(x) = x^2 - 1, g(x) = x + 2 (f / g)(2) = f(2) / g(2) = 3 / 4 (f / g)(x) = f(x) / g(x) = x^2 - 1 / x + 2, x not = -2

Product of two functions

(fg)(x) = f(x) • g(x) f(x) = x^3, g(x) = 1 - 3x (fg)(-2) = f(-2) • g(-2) = (-8)(7) = -56 (fg)(x) = f(x) • g(x) = x^3(1 - 3x) = x^3 - 3x^4

Finding intercepts

1. To find x-intercepts, let y = 0 and solve for x. 2. To find y-intercepts, let x = 0 and solve for y. 1. In 3x + 5y = 15 let y = 0 to obtain 3x = 15, or x = 5. The x-intercept is 5. 2. In 3x + 5y = 15 let x = 0 to obtain 5y = 15, or y = 3. The y-intercept is 3.

Function

A function is a relation in which each valid input results in one output. The domain of a function is the set of valid inputs (x-values), and the range is the set of resulting outputs (y-values). f = {(1, 3), (2, 6), (3, 9), (4, 9)} The domain is D = {1, 2, 3, 4}, and the range is r = {3, 6, 9}.

Nonlinear function

A nonlinear function cannot be expressed in the form f(x) = mx + b. f(x) = root x + 1, g(x) = 4xcubed, and h(x) = x^1.01 + 2.

Relation, domain, and range

A relation is a set of ordered pairs (x, y). The set of x-values is called the domain, and the set of y-values is called the range. The relation S = {(1,3), (2,5), (1,6)} has domain D = {1,2} and range R = {3, 5, 6}.

Slope-intercept form y = mx + b

A unique equation for a line, determined by the slope m and the y-intercept b. An equation of the line with slope 5 and y-intercept -4 is y = 5x - 4.

Interval notation

An efficient notation for writing inequalities. x < or = 6 is equivalent to (-infinity, 6]. x > 3 is equivalent to (3, infinity). 2 < x < or = 5 is equivalent to (2, 5].

Real numbers

Any number that can be expressed in standard (decimal) form Include the rational numbers and irrational numbers pi, root 7, -4/7, 0, -10, 1.237 0.6 cont. = 2/3, 1000, root 15, -root 5

Difference quotient

Calculates average rate of change of f from x to x + h. f(x + h) - f(x) / h, h not = 0 If f(x) = 2x, then the difference quotient equals 2(x + h) - 2x / h = 2h / h = 2.

Irrational numbers

Can be written as nonrepeating, nonterminating decimals; cannot be a rational number; if a square root of a positive integer is not an integer, it is an irrational number pi, root 2, -root 5, ^3 root 7, pi^4

Graphical representation of a function

Graph of ordered pairs (x, y) that satisfies y = f(x). Each point on the graph satisfies y = #x.

Average rate of change of f from x1 to x2

If (x1, y1) and (x2, y2) are distinct points on the graph of f, then the average rate of change from x1 to x2 equals y2 - y1 / x2 - x1. If f(x) = 3xsquared, then the average rate of change from x = 1 to x = 3 is 27 - 3 / 3 - 1 = 12 because f(3) = 27 and f(1) = 3. This means that, on average, f(x) increases by 12 units for each unit increases in x from 1 to 3.

Inverse function

If a function f is one-to-one, it has an inverse function f^-1 that satisfies both (f^-1of)(x) = f^-1(f(x)) = x and (fof^-1)(x) = f(f^-1(x)) = x. f(x) = 3x - 1 is one-to-one and has inverse function f^-1(x) = x + 1 / 3. f^-1(f(x)) = f^-1(3x - 1) = (3x - 1) + 1 / 3 = x Similarly, f(f^-1(x)) = x.

Rational numbers

Include integers; all fractions p/q, where p and q are integers with q not = 0; all repeating and all terminating decimals 1/2, -3, 128/6, -0.335, 0, 0.25 = 1/4, 0.33 cont. = 1/3

Symbolic representation of a function

Mathematical formula. The squaring function is given by f(x) = xsquared, and the square root function is given by g(x) = root x

One-variable data

Number line, list, one-column or one-row table The data items are the same type and can be described using x-values. Computations of the mean and median are performed on one-variable data.

Linear function

Slope of graph: Always constant Graph: Nonvertical line

Constant function

Slope of graph: Always zero. Graph: Horizontal line

Nonlinear function:

Slope of graph: No notion of one slope Graph: Not a line

Natural numbers

Sometimes referred to as the "counting numbers" 1, 2, 3, 4, 5,...

Standard equation of a circle

The circle center (h, k) and radius r has the equation (x - h)squared + (y - k)squared = rsquared

Distance formula

The distance between (x1, y1) and (x2, y2) is d = root over all (x2 - x1)squared + (y2 - y1)squared. The distance between (2, -1) and (-1, 3) is d = root over all (-1 - 2)squared + (3 - (-1))squared = 5.

Domains and ranges of inverse functions

The domain of f equals the range of f^-1. The range of f equals the domain of f^-1. Let f(x) = (x + 2)^2 with restricted domain x > or = -2 and range y > or = 0. It follows that f^-1(x) = root x - 2 with domain x > or = and range y > or = -2.

Least-squares regression line

The line of least-squares fit for the points (1, 3), (2,5), and (3, 6) is y = 3 / 2 x + 5 / 3 and r about = 0.98. Try verifying this with a calculator.

Median

The median of a sorted list of numbers equals the value that is located in the middle of the list. Half the data are greater than or equal to the median, and half the data are less than or equal to the median. The median of 2, 3, 6, 9, 11 is 6, the middle data item. The median of 2, 3, 6, 9 is the average of the two middle values: 3 and 6. Therefore the median is: 3 + 6 / 2 = 4.5.

Midpoint formula

The midpoint of the line segment connecting (x1, y1) and (x2, y2) is = (x1 + x2 / 2, y1 + y2 / 2). The midpoint of the line segment connecting (4, 3) and (-2, 5) is M = (4 + (-2) / 2, 3 + 5 / 2) = (1, 4).

Correlation coefficient r

The values of r satisfy -1 < or = r < or =, where a line fits the data better if r is near -1 or 1. A value near 0 indicates a poor fit.

Mean, or average

To find the mean, or average, of n numbers, divide their sum by n. The mean of the four numbers -3, 5, 6, 9 is -3 + 5 + 6 + 9 / 4 = 4.25.

Two-variable data

Two-column or two-row table, scatterplot, line graph or other type of graph in the xy-plane. Two types of data are related, can be described by using ordered pairs (x, y), and are often called a relation.

Point-slope form y = m(x - x1) + y1 or y - y1 = m(x - x1)

Used to find the equation of a line, given two points or one point and the slope. Given two points (5, 1) and (4, 3), first compute m = 3 - 1 / 4 - 5 = -2. An equation of this line is y = -2(x - 5) + 1.

Order of operations

Using the following order of operations, perform all calculations within parentheses, square roots, and absolute value bars and above and below fraction bars. Then perform any remaining calculations. 1. Evaluate all exponents. Then do any negation after evaluating exponents. 2. Do all multiplication and division from left to right. 3. Do all addition and subtraction from left to right. -4squared minus 12 divide 2 = -16 - 12 divide 2 - 2 = -16 - 6 - 2 = -22 - 2 = -24 2 + 4squared / 3 - 3 time 5 = 2 + 16 / 3 - 15 = 18 / -12 = -3 / 2

Identifying graphs of functions

Vertical Line Test: if every vertical line intersects a graph at no more than one point, then the graph represents a function. (Otherwise the graph does not represent a function.)

Implied domain

When a function is represented by a formula, it's domain, unless otherwise stated, is the set of all valid inputs (x-values) that are defined or make sense in the formula. f(x) = 1 / x + 4 Domain of f: {x| x not = -4}

Verbal representation of a function

Words describe precisely what is computed. A verbal representation of f(x) = xsquared is "Square the input x to obtain the output."

Increasing and decreasing

f increases on an open interval if, whenever x1 < x2, then f(x1) < f(x2). f decreases on an open interval if, whenever x1 < x2, then f(x1) > f(x2).

One-to-one function

f is one-to-one if different inputs always result in different outputs. That is, a not = b implies f(a) not = f(b). f(x) = x^2 - 4x is not one-to-one because f(0) = 0 and f(4) = 0. With this function, different inputs can result in the same output.

Constant function

f(x) = b, where b is a fixed number, or constant. f(x) = 12, g(x) = -2.5, and h(x) = 0. Every constant function is also linear.

Linear function

f(x) = mx + b, where m and b are constants. The graph of f has slope m. f(x) = 3x - 1, g(x) = -5, and h(x) = 1 / 2 - 3 / 4 x. Their graphs have slopes 3, 0, and -3 / 4.

Slope of a line passing through (x1, y1) and (x2, y2)

m = delta y / delta x = y2 - y1 / x2 - x1 Delta y = y2 - y1 denotes the change in y. Delta x = x2 - x1 denotes the change in x. A line passing through (-1, 3) and (1, 7) has slope m = 7 - 3 / 1 - (-1) = 4 / 2 = 2. This slope indicates that the line rises 2 units for each unit increase in x.

Vertical line

x = k, where k is a constant. A vertical line with x-intercept -8 has the equation x = -8.

Horizontal line

y = b, where b is a constant. A horizontal line with y-intercept 7 has the equation 7 = 7.

Parallel lines

y = m1x + b1 and y = m2x + b2, where m1 = m2. The liens given by y = -3x - 1 and y = -3x + 5 are parallel because they both have slope -3.

Perpendicular lines

y = m1x + b1 and y = m2x + b2, where m1m2 = -1. The lines y = 2x - 5 and y = -1 / 2 x + 2 are perpendicular because m1m2 = 2(-1 / 2) = -1.