PreCalculus 3-R Unit 3 Functions and Graphs. Review Problems Evaluate the function f (x) = x 2 + 6x at f (8). f (8) = 112 Evaluate the function at f (6).

Slides:

Advertisements

Similar presentations

1.6 Inverse Functions Students will find inverse functions informally and verify that two functions are inverse functions of each other. Students will.

Advertisements

Operations on Functions Composite Function:Combining a function within another function. Written as follows: Operations Notation: Sum: Difference: Product:

COMPOSITE AND INVERSE FUNCTIONS Mrs. Aldous, Mr. Beetz & Mr. Thauvette IB DP SL Mathematics.

Section 1.2 Basics of Functions

6.5 - Graphing Square Root and Cube Root

12.1 Inverse Functions For an inverse function to exist, the function must be one-to-one. One-to-one function – each x-value corresponds to only one y-value.

Exponential and Logarithmic Functions and Equations

(1) What is the slope of the line through (2,3) and (5,-9)? A. 2 B. 0 C.-4 D.-1.

A. What is the definition of Domain? B. What is the definition of Range? Your answers should be a complete sentence.

Functions Domain and range The domain of a function f(x) is the set of all possible x values. (the input values) The range of a function f(x) is the set.

2 pt 3 pt 4 pt 5 pt 1 pt 2 pt 3 pt 4 pt 5 pt 1 pt 2 pt 3 pt 4 pt 5 pt 1 pt 2 pt 3 pt 4 pt 5 pt 1 pt 2 pt 3 pt 4 pt 5 pt 1 pt Functions Quadratics 1 Quadratics.

1 PRECALCULUS I Dr. Claude S. Moore Danville Community College Composite and Inverse Functions Translation, combination, composite Inverse, vertical/horizontal.

Unit 6 GA2 Test Review. Find the indicated real n th root ( s ) of a. a. n = 3, a = –216 b. n = 4, a = 81 SOLUTION b. Because n = 4 is even and a = 81.

Functions Review.

Combinations of Functions & Inverse Functions Obj: Be able to work with combinations/compositions of functions. Be able to find inverse functions. TS:

DO NOW: 6.3: w/s C: Perform the indicated operation. 1.) Find g(f(x)) if f(x) = 2x 2 – x and g(x) = 2.) Find g(h(8)) if g(x) = -x 2 and h(x) =

Jeopardy Growing Exponentially Welcome to the Function Fun with Functions Snoitcunf A special Function Let’s Change it Up

PRECALCULUS Inverse Relations and Functions. If two relations or functions are inverses, one relation contains the point (x, y) and the other relation.

Piecewise Graphs A piecewise function is defined by at least two equations, each of which applies to a different part of the function’s domain. One example.

5.1 Composite Functions Goals 1.Form f(g(x)) = (f  g) (x) 2.Show that 2 Composites are Equal.

Composite Functions How would you define composite functions? Math30-1.

Logarithmic Functions & Their Graphs

More Quarter test review Section 4.1 Composite Functions.

Vertical and Horizontal Shifts of Graphs.  Identify the basic function with a graph as below:

IFDOES F(X) HAVE AN INVERSE THAT IS A FUNCTION? Find the inverse of f(x) and state its domain.

Copyright © Cengage Learning. All rights reserved. Pre-Calculus Honors 1.3: Graphs of Functions HW: p.37 (8, 12, 14, all, even, even)

Jeopardy Math Review Evaluating a Function Examining f(x) Examining the Graph of f(x) Combining Functions Inverses

Do Now: Find f(g(x)) and g(f(x)). f(x) = x + 4, g(x) = x f(x) = x + 4, g(x) = x

1.4 Building Functions from Functions

Transformation of Functions Sec. 1.7 Objective You will learn how to identify and graph transformations.

Function Notation Assignment. 1.Given f(x) = 6x+2, what is f(3)? Write down the following problem and use your calculator in order to answer the question.

1.1B Horizontal and Vertical Translations

Chapter 7 – Radical Equations and Inequalities 7.2 – Inverse Functions and Relations.

Practice #9 p eoo Findwhe n. Inverse functions - one function undoes the other. x f(x) x g(x) Definition of.

Inverse Functions The inverse of a function is obtained by interchanging the x and y values of the original function. Inverse Function Notation: If the.

6.4 Notes – Use Inverse Functions. Inverse: Flips the domain and range values Reflects the graph in y = x line. Functions f and g are inverses of each.

5.3 Inverse Functions. Definition of Inverse Function A function of “g” is the inverse function of the function “f” if: f(g(x)) = x for each x in the.

1.8 Inverse Functions. Any function can be represented by a set of ordered pairs. For example: f(x) = x + 5 → goes from the set A = {1, 2, 3, 4} to the.

Bellwork: Match the following graphs with their equations. NO CALCS! y=-x 2 X=# x=y 2 x = -y

6.2 Inverse functions and Relations 1. 2 Recall that a relation is a set of ordered pairs. The inverse relation is the set of ordered pairs obtained by.

Ch 9 – Properties and Attributes of Functions 9.5 – Functions and their Inverses.

Today in Pre-Calculus No calculators needed Notes: –Rational Functions and Equations –Transformations of the reciprocal function Go over quiz Homework.

Do Now: Given f(x) = 2x + 8 and g(x) = 3x 2 – 1 find the following. 1.) (f + g)(x) 2.) g(x – 2)

5.3 Inverse Functions (Part I). Objectives Verify that one function is the inverse function of another function. Determine whether a function has an inverse.

Inverse Functions Pages in Text Any Relevant Graphics or Videos.

Warm up 1. Graph the following piecewise function:

Objectives: 1)Students will be able to find the inverse of a function or relation. 2)Students will be able to determine whether two functions or relations.

Calculus P - Review Review Problems

Copyright © 2011 Pearson Education, Inc. Logarithmic Functions and Their Applications Section 4.2 Exponential and Logarithmic Functions.

Objectives: To find inverse functions graphically & algebraically.

Unit 3 Functions and Graphs

Functions and Their Graphs RAFIZAH KECHIL, UiTM PULAU PINANG

DO NOW: Perform the indicated operation.

Basic Math Skills.

Chapter 2: Analysis of Graphs of Functions

Functions Review.

A function is given by a formula. Determine whether it is one-to-one

Warm-up: Solve for x. 2x = 8 2) 4x = 1 3) ex = e 4) 10x = 0.1

True or False: Suppose the graph of f is given

Piecewise Functions At least 2 equations, each of which applies to a different part of the functions domain. It is like having 3 equations for 3 different.

Graphing and Evaluating The Piecewise Function A Series of Examples

Composition of Functions And Inverse Functions.

1.2 Analyzing Graphs of Functions and Relations

§ 8.3 Graphing Piecewise-Defined Functions and Shifting and Reflecting Graphs of Functions.

Determine if 2 Functions are Inverses by Compositions

Horizontal and Vertical Translations

Warm Up Determine the domain of f(g(x)). f(x) = g(x) =

Review for Test #1 Calculus – Larson.

Presentation transcript:

PreCalculus 3-R Unit 3 Functions and Graphs

Review Problems Evaluate the function f (x) = x 2 + 6x at f (8). f (8) = 112 Evaluate the function at f (6). f (6) = 28 1

Review Problems Evaluate the following piecewise defined function at f (1), f (3), and f (7). f (1) = 1, f (3) = 18, f (7) = 50 Determine whether the equation defines y as a function of x Yes 2

Review Problems Use the function f (x) = x to evaluate the following expressions and simplify. f (x + 5) and f (x) + f (5) For the function, find. 3

Review Problems What is the domain of the function  What is the domain of the function 4

Review Problems Sketch the graph of the piecewise defined function 5

Review Problems Sketch the graph of the piecewise defined function 6

Review Problems Sketch the graph of the piecewise defined function 7

Review Problems Determine whether the equation defines y as a function of x. No Consider a family of functions. How does the value of c affect the graph? The graphs are obtained by shifting the graph of upward c units, 8

Review Problems The graph of g is given. Sketch the graph of the function 9

Review Problems The function f (x) is reflected in the x-axis and then shifted up 5 units and the graph of g (x) = 5 – x 2 is obtained. What is f (x)? f (x) = x 2 10

Review Problems The graph of f is given. Sketch the graph of the function y = –f(x)

Review Problems Express the function in the form Find the inverse function of 12

Review Problems Use the given graphs of f and g to evaluate g (f (5)). 3 13

Review Problems For find Find the inverse function of 14

Review Problems A function f is given. Sketch the graph of f. Use the graph of f to sketch the graph of. Find 15

Review Problems Assume f is a one-to-one function. If f (x) = 3 – 6x, find f –1 (33). –5 Use the Property of Inverse Functions to find the inverse function of f(x) = x + 8 f –1 (x) = x – 8 16

Review Problems Find the inverse function of 17

Review Problems Express the function in the form evaluate f(g(–1))

Review Problems Use the given graphs of f and g to evaluate g (f (5)) 3 19

Review Problems Find the domain of Suppose that g(x) = 5x + 3 and h(x) = 25x x Find a function f, such that f(g(x)) = h(x). f(x) = x

Review Problems A function f is given Sketch the graph of f. Use the graph of f to sketch the graph of. Find 21

Review Problems A one-to-one function is given Find the inverse of the function. Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line y=x. 22

Review Problems A one-to-one function is given Find the inverse of the function. Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line y=x. 23

Review Problems Assume f is a one-to-one function. If f (x) = 3 – 6x, find f –1 (33). -5 Use the Property of Inverse Functions to find the inverse function of f(x) = x + 8. f –1 (x) = x – 8 24

Review Problems Find the inverse function of 25

Review Problems 26

Review Problems 27

Review Problems 28

Review Problems 29

Review Problems 30

Review Problems 31

Review Problems 32

Review Problems 33

Review Problems 34

Answers f (8) = 112 f (6) = 28 f (1) = 1, f (3) = 18, f (7) = 50 Yes No The graphs are obtained by shifting the graph of upward c units, f (x) = x 2 3

Answers –5 f –1 (x) = x – f(x) = x f –1 (x) = x – 8

Answers