Math: Calculus [1] Functions and Models

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absolute value

For a number a, this term, denoted by | a |, is the distance from a to 0 on the real number line.

step functions

Functions similar to figure 18 are called ________.

composite function (composition)

Given two functions f and g, the ________ function, f of g, is defined by (f ° g)(x) = f(g(x))

even function

If a function f satisfies f(x) = f(-x) for every number x in its domain, then f is called an ________.

shrink the graph vertically

If c > 1, how will y = (1/c)*f(x) transform the graph of y = f(x)?

reflect the graph about the x-axis

If c > 1, how will y = -f(x) transform the graph of y = f(x)?

stretch the graph vertically

If c > 1, how will y = c*f(x) transform the graph of y = f(x)?

reflect the graph about the y-axis

If c > 1, how will y = f(-x) transform the graph of y = f(x)?

shrink the graph horizontally

If c > 1, how will y = f(x*c) transform the graph of y = f(x)?

stretch the graph horizontally

If c > 1, how will y = f(x/c) transform the graph of y = f(x)?

odd function

If f satisfies f(-x) = -f(x) for every number x in its domain, then f is called an ________.

inverse function

Let f be a one-to-one function with domain A and range B. Then its ________ has domain B and range A and is defined (by the figure) for any y in B.

to the left

Suppose c > 0. To obtain the graph of y = f(x + c), shift the graph of y = f(x) a distance c units ________.

to the right

Suppose c > 0. To obtain the graph of y = f(x - c), shift the graph of y = f(x) a distance c units ________.

upward

Suppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units ________.

downward

Suppose c > 0. To obtain the graph of y = f(x) - c, shift the graph of y = f(x) a distance c units ________.

laws of logarithms

Taken together, the following equations are referred to as ________.

law of exponents

Taken together, the following equations are referred to as the ________.

cancellation equations

The following equations are referred to as ________.

logarithmic function with base b

The following equations represent what type of function?

graph

The following figure represents a ________ for a function f.

machine diagram

The following figure represents a ________ for a function f.

arrow diagram

The following figure represents an ________ for a function f.

The natural log of e is equal to what integer?

the vertical line test

This term is described by the following definition: A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once.

family of functions

This term refers to a collection of functions whose equations are related.

reciprocal function

This term refers to a function of the form f(x) = 1/x

power function

This term refers to a function of the form f(x) = x^a , where a is a constant.

root function

This term refers to a function of the form f(x) = ⁿ√x , where n is a constant.

linear function

This term refers to a function whose graph is a straight line.

natural logarithm

This term refers to a logarithm with base e

mathematical model

This term refers to a mathematical description (often by means of a function or an equation) of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions.

empirical model

This term refers to a model which is based entirely on collected data.

coefficient

This term refers to a numerical or constant quantity placed before and multiplying the variable in an algebraic expression (e.g., 4 in 4x).

quadratic function

This term refers to a polynomial function, in one or more variables, in which the highest-degree term is of the second degree.

cubic function

This term refers to a polynomial function, in one or more variables, in which the highest-degree term is of the third degree.

rational function

This term refers to a ratio of two polynomials.

function

This term refers to a rule that assigns to each element x in a set D exactly one element, called f(x), in a set E.

translations

This term refers to a type of transformation that shifts a graph upward, downward, left, or right.

stretching

This term refers to a type of transformation that stretches (or shrinks) the values of each point in a figure by a constant factor.

reflecting

This term refers to a type of transformation that takes each point in a figure and reflects it over a line.

polynomial

This term refers to an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

natural exponential function

This term refers to f(x)=e^x

piecewise defined functions

This term refers to functions defined by different formulas in different parts of their domains.

trigonometric functions

This term refers to functions of an angle. They relate the angles of a triangle to the lengths of its sides. These functions are important in the study of triangles and modeling periodic phenomena, among many other applications.

exponential functions

This term refers to functions of the form f(x) = b^x , where the base (b) is a positive constant.

logarithmic functions

This term refers to functions of the form f(x) = log_b(x), where the base (b) is a positive constant.

algebraic functions

This term refers to functions that can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) starting with polynomials.

degree

This term refers to the class into which an equation falls according to the highest power of unknowns or variables present.

horizontal line test

This term refers to the following definition: a function is one-to-one if, and only if, no horizontal line intersects its graph more than once.

(1) verbally,(2) numerically,(3) visually, and (4) algebraically

What are the four possible ways to represent a function?

log_b(x) = y

What equation is equivalent to b^y = x

b^y = x

What equation is equivalent to log_b(x) = y

y = f(x/c)

What equation would best represent the transformation of y = f(x) from red to blue? (using a single constant, c)

y = -f(x)

What equation would best represent the transformation of y = f(x) from red to green?

y = c*f(x)

What equation would best represent the transformation of y = f(x) from red to green? (using a single constant, c)

y = f(-x)

What equation would best represent the transformation of y = f(x) from red to orange?

y = (1/c)*f(x)

What equation would best represent the transformation of y = f(x) from red to orange? (using a single constant, c)

y = f(x*c)

What equation would best represent the transformation of y = f(x) from red to violet? (using a single constant, c)

e^y = x

What equation would complete the identity?

ln(x) = y

What equation would complete the identity?

y = f(x) - c

What equation would represent the graph of y = f(x) translated c units downward?

y = f(x + c)

What equation would represent the graph of y = f(x) translated c units to the left?

y = f(x - c)

What equation would represent the graph of y = f(x) translated c units to the right?

y = f(x) + c

What equation would represent the graph of y = f(x) translated c units upward?

(a^x)(b^x)