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The French Revolution

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MAT.126 1.4-1.5

luonut David Kedrowski

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Introduction to Operations on Functions

Evaluating a function means finding the value of f(x) =… or y =… that corresponds to a given value of x. To do this, simply replace all the x variables with whatever x has been assigned. For example, if we are asked to evaluate f(4), then x has been assigned the value of 4

The Sum of Two Function

(f + g)(x) = f(x) + g(x), where x is in the domain of both f and g. = 2x3 + x2 + 2. The domain of (f + g)(x) consists of all x-values that are in the domain of both f and g.

To find the domain of a function, just plug the x-values into the quadratic formula to get the y-output. To find the range of a function, first find the x-value and y-value of the vertex using the formula x = -b/2a. Then, plug that answer into the function to find the range.

How do you find the domain of each function?

Identify the input values. Identify any restrictions on the input. If there is a denominator in the function's formula, set the denominator equal to zero and solve for x . ... Write the domain in interval form, making sure to exclude any restricted values from the domain.

Find the domain of g. Find the domain of f. Find those inputs x in the domain of g for which g(x) is in the domain of f. That is, exclude those inputs x from the domain of g for which g(x) is not in the domain of f. The resulting set is the domain of f∘g.

Finding a Difference Quotient For a Linear Function

Given a function f(x), and two input values, x and x + h (where h is the distance between x and x + h), the difference quotient is the quotient of the difference of the function values, f(x + h) - f(x), and the difference of the input values, (x + h) - x.

the difference quotient formula is nothing but the slope of a secant line formula. The difference quotient of a function y = f(x) is given by [ f(x + h) - f(x) ] / h.

The difference quotient for f is defined as f(x+h)−f(x)h. This represents the slope of the secant line of the curve y=f(x), through the points (x,f(x)) and (x+h,f(x+h)).

Composing Functions

n mathematics, function composition is an operation ∘ that takes two functions f and g, and produces a function h = g ∘ f such that h(x) = g(f(x)). In this operation, the function g is applied to the result of applying the function f to x.

Finding a Difference Quotient For a Nonlinear Function

Plug x + h into the function f and simplify to find f(x + h). Now that you have f(x + h), find f(x + h) - f(x) by plugging in f(x + h) and f(x) and simplifying. Plug your result from step 2 in for the numerator in the difference quotient and simplify.

Introduction to Difference Quotient

The difference quotient is used to calculate the slope of the secant line between two points on the graph of a function, f. Just to review, a function is a line or curve that has only one y value for every x value. It's like an input/output machine.

Operations on functions are those that can be performed on functions, such as arithmetic operations. The most common operation is addition, but subtraction, multiplication, and division are also possible. Other operations include composition, inversion, and evaluation.

Solving Quadratic Inequalities

The quadratic equation ax2 + bx + c = 0 is written as a quadratic equation by replacing equals to the symbol (=) with greater than or lesser than inequality. The general format of a quadratic inequality is ax2 + bx + c > 0, or ax2 + bx + c < 0.

Solving a Polynomial Inequality with Special Case Solution Sets

A quadratic polynomial is a second-degree polynomial where the value of the highest degree term is equal to 2. The general form of a quadratic equation is given as ax2 + bx + c = 0. Here, a and b are coefficients, x is the unknown variable and c is the constant term.

A polynomial function with degree greater than 2 has a graph that is some type of curve. The graph of a polynomial function will usually partially lie above the x-axis and partially lie below the x-axis. The locations (values of x) where it crosses or touches the x-axis are called zeros.

Introduction to Polynomial Inequalities

a polynomial inequality is expressed with the polynomial on one side and a zero on the other side. The solution to a polynomial inequality will always be a set or interval of numbers instead of a single number. The solution is a range of numbers that all make the inequality true.

A polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using mathematical operations such as addition, subtraction, multiplication and division (No division operation by a variable).

When solving an inequality: • you can add the same quantity to each side • you can subtract the same quantity from each side • you can multiply or divide each side by the same positive quantity If you multiply or divide each side by a negative quantity, the inequality symbol must be reversed.

Equations and inequalities are both mathematical sentences formed by relating two expressions to each other. In an equation, the two expressions are deemed equal which is shown by the symbol =. Where as in an inequality, the two expressions are not necessarily equal which is indicated by the symbols: >, <, ≤ or ≥

Introduction to Rational Inequalities

A rational inequality is an inequality that contains a rational expression, where a rational expression is a ratio of two polynomials. That is, a rational expression is of the form R(x) / Q(x), where R(x) and Q(x) are polynomials and Q(x) is not zero.

Introduction to Rational Functions

A rational expression is simply a quotient of two polynomials. Or in other words, it is a fraction whose numerator and denominator are polynomials.

Rational expressions and undefined values Consider the rational expression

We can determine the value of this expression for particular x-values.

Introduction to Vertical and Horizontal Asymptotes

Subtopic

A vertical asymptote of a graph is a vertical line x = a where the graph tends toward positive or negative infinity as the inputs approach a. A horizontal asymptote of a graph is a horizontal line y = b where the graph approaches the line as the inputs approach ∞ or –∞.

identifying Vertical Asymptotes Algebraically

Vertical Asymptotes. If the limit of f(x) as x approaches c from either the left or right (or both) is ∞ or −∞, we say the function has a vertical asymptote at c

The vertical asymptote will always occur where the denominator is equal to zero. Set the denominator equal to zero and solve for x: x² - 3x - 2 = 0. (x-2) (x-1) = 0.

Using the Degree of the Numerator and Denominator to Determine Horizontal Asymptotes

A function of the form f(x) = a (bx) + c always has a horizontal asymptote at y = c. For example, the horizontal asymptote of y = 30e–6x – 4 is: y = -4, and the horizontal asymptote of y = 5 (2x) is y = 0.

If the numerator's degree is less than the denominator's degree, then the horizontal asymptote is y = 0. If the numerator's degree is equal to the denominator's degree, then the horizontal asymptote is y = c, where c is the ratio of the leading terms or their coefficients.

If the top degree is greater than the bottom degree. There is actually no horizontal asymptote.

vertical asymptote as x = - d c . If the degrees of the numerator and denominator are the same, the horizontal asymptote is y = a c .

Identifying Horizontal Asymptotes Algebraically

If n < m, the horizontal asymptote is y = 0. If n = m, the horizontal asymptote is y = a/b. If n > m, there is no horizontal asymptote.

A horizontal asymptote is found by comparing the leading term in the numerator to the leading term in the denominator

dividing the numerator of the function rule by the denominator and using the first two terms in the quotient in the equation of the line that is the asymptote

The horizontal asymptote of a rational function can be determined by looking at the degrees of the numerator and denominator. If N is the degree of the numerator and D is the degree of the denominator, and… N < D, then the horizontal asymptote is y = 0.

To find the vertical asymptotes, set the denominator equal to zero and solve for x. This is already factored, so set each factor to zero and solve. Since the asymptotes are lines, they are written as equations of lines.

A polynomial is an expression that consists of a sum of terms containing integer powers of x like 3x2-6x-1

A rational function is a function which is the ratio of polynomial functions. Said differently, r is a rational function if it is of the for

The line y=c is called a horizontal asymptote of the graph of a function

the line x=c is called a vertical asymptote

Introduction to Polynomial Functions

key words

Multiplicity

Zero

Polynomial

Introduction to the Intermediate Value Theorem

Discover the Intermediate Value Theorem, a fundamental concept in calculus that states if a function is continuous over a closed interval [a, b], it encompasses every value between f(a) and f(b) within that range

Applying the Intermediate Value Theorem

Define a function y=f(x). Define a number (y-value) m. Establish that f is continuous. Choose an interval [a,b]. Establish that m is between f(a) and f(b). Now invoke the conclusion of the Intermediate Value Theorem.

The intermediate value theorem has many applications. Mathematically, it is used in many areas. This theorem is utilized to prove that there exists a point below or above a given particular line. It is also used to analyze the continuity of a function that is continuous or not.

The conditions that must be satisfied in order to use Intermediate Value Theorem include that the function must be continuous and the number must be within the interva

Turning Points of a Graph of a Polynomial Function

A turning point of a function is a point where the graph of the function changes from sloping downwards to upwards, or vice versa. The gradient changes from positive to negative or negative to positive. A polynomial of degree n can have up to (n−1) turning points.

The general (polynomial) form: y = ax2 + bx + c. The turning point form: y = a(x - h)2 + k. Factorised form: y = a(x - m)(x - n)

To flip or reflect (vertically) about the horizontal x-axis, replace y = f(x) with y = -f(x).

Identifying the End Behavior of the Graph of a Polynomial Function

The highest degree of polynomial equations determine the end behavior. -- If the degree is even, like y=x^2; y=X^4; y=x^6; etc., then the ends will extend in the same direction. -- If the degree is odd, like y=x^3; y=x^5; y=x^7; etc., then the ends will move in opposite directions.

Determining Zeros and Multiplicities

A zero of a polynomial is an x-value for which the polynomial equals zero. This means that if x = c is a zero, then . The zeros correspond to the x-intercepts of the polynomial.

The multiplicity of a zero, x = c, is the number of times the factor appears in the fully factored form of the polynomial. This is equal to the exponent on the factor.

We will use these steps, definitions, and equations to find zeros and their multiplicities given a polynomial function written in factored form in the following two examples.

Identifying Zeros and Multiplicities

A function is a relation where each input value (x-value) has only one output (y-value). Thus, all functions are relations. But, not all relations are functions because not all will meet the requirement that each unique input creates only one output

Notation for Infinite Behavior

The end behavior of a polynomial function is the behavior of the graph of as approaches positive infinity or negative infinity. The degree and the leading coefficient of a polynomial function determine the end behavior of the graph. The leading coefficient is significant compared to the other coefficients in the function for the very large or very small numbers. So, the sign of the leading coefficient is sufficient to predict the end behavior of the function.

Introduction to the Leading Term Test

The Leading-Term Test. The theory states: If anxn is the leading term of a polynomial function, then the behavior of the graph as x →∞ and as. x →-∞ can be described in one of the four following ways: Degree Leading.