Basics of functions continued
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This document discusses evaluating functions using different representations including graphs, tables, and equations. It provides examples of evaluating composite functions involving addition, subtraction, multiplication, and division using the different representations. For graphs, it demonstrates evaluating composite functions by looking at the y-values of the individual functions when x is the given value. For tables, it shows evaluating composite functions by looking up the function values in the tables when x is the given value and then performing the indicated operation.
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- Sets can contain numbers, elements, and be represented using curly brackets.
- Venn diagrams use overlapping circles to show logical connections between sets.
- A function has a domain (input) and range (output), where each input is mapped to a unique output.
- Composite functions combine other functions by substituting one into another.
- Inverse functions reverse the input and output of a function if it exists.
- Common functions that can be graphed include linear, quadratic, trigonometric, cubic, exponential and logarithmic functions.
Invers fungsi
The document involves calculating inverse functions of various compositions of functions f(x) and g(x). It provides the definitions of f(x) and g(x) and calculates:
1) The inverse functions f^-1(x) and g^-1(x)
2) The compositions (f o g)(x), (g o f)(x), and their inverse functions
3) The inverse functions of (f o g)^-1(x) and (g o f)^-1(x)
4) Confirms the relationship that (f o g)^-1(x) = (g^-1 o f^-1)(x) and (g o f)^-1
52 notation and algebra of functions
The document discusses notation and algebra of functions. It defines functions as procedures that assign a unique output to each valid input. Most functions are given by formulas like f(x) = x^2 - 2x + 3, where the input is placed in parentheses and evaluated to get the output. Expressions can be formed by using function notation and evaluating one function's output as the input to another function. An example evaluates two functions f(x) = -3x + 2 and g(x) = -2x^2 - 3x + 1 at different inputs and combines them into one expression.
Derivatives in Multi
1) The document discusses the generalization of derivatives from single-variable calculus to multivariable calculus. In multivariable calculus, a function f(x,y) is differentiable at a point (a,b) if the graph near that point is indistinguishable from a plane.
2) The derivative of a function of two variables f(x,y) at a point (a,b) is represented by the Jacobian matrix Jf(a,b), which can be written as the gradient vector grad f(a,b).
3) The directional derivative of f(x,y) at (a,b) in the direction of a unit vector u is
F4 c1 functions__new__1_
1. The relation between sets A and B is many-to-one. The range of the relation is {p, q, r}.
2. The images of a are b and d. The object of b is f and h.
3. The value of k is 2. Using function notation, g(x) = x - 2.
Algebra 2. 9.15. Intro to quadratics
The document provides information about graphing and transforming quadratic functions:
- It discusses graphing quadratic functions by making tables of x- and y-values and plotting points. Examples show graphing f(x) = x^2 - 4x + 3 and g(x) = -x^2 + 6x - 8.
- Transformations of quadratic functions are described as translating the graph left/right or up/down, reflecting across an axis, or stretching/compressing vertically or horizontally. Examples demonstrate translating, reflecting, and compressing the graph of f(x) = x^2.
- The vertex form of a quadratic function f(x) = a(x-h)^2 +
Alg 1 ch. 5.3 - functions
The document discusses functions and relations. It defines a relation as a set of ordered pairs and shows an example relation. It explains that a relation is a function if each x-value is mapped to only one y-value using the vertical line test. The document provides examples of function notation, evaluating functions, and graphing functions from equations. It includes examples of making tables of values and ordered pairs to graph functions.
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Basics of functions continued
- 2. Domain