Parametric and Inverse Functions Nate Hutnik Aidan Lindvig Craig Freeh.

Slides:

Advertisements

Similar presentations

6.7 Notes – Inverse Functions. Notice how the x-y values are reversed for the original function and the reflected functions.

Advertisements

6.2 One-to-One Functions; Inverse Functions

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

Example 1A: Using the Horizontal-Line Test

Verifying Inverses Verifying –checking that something is true or correct Inverses – two equations where the x’s and y’s of one equation are switched in.

1.4c Inverse Relations and Inverse Functions

Solve Systems of Equations by Elimination

7.4 Inverse Functions p Review from chapter 2 Relation – a mapping of input values (x-values) onto output values (y-values). Here are 3 ways to.

Precalculus 1.7 INVERSE FUNCTIONS.

Functions and Their Inverses

Logarithmic Functions  In this section, another type of function will be studied called the logarithmic function. There is a close connection between.

Algebra 2: Section 7.4 Inverse Functions.

Sit in the same seat as yesterday

Inverses Graph {(1,2), (2,-4), (5,2), (2, 0), (-1, 4)} ? {(2, 1), (-4, 2), (2, 5), (0,2), (4, -1)} These graphs are called inverses of each other. Graph.

Objectives Determine whether the inverse of a function is a function.

Functions and Their Inverses

Operations with Functions Section 2.4.  Sum  Difference  Product  Quotient  Composition Types of Operations.

Math-3 Lesson 4-1 Inverse Functions. Definition A function is a set of ordered pairs with no two first elements alike. – f(x) = { (x,y) : (3, 2), (1,

Copyright © 2011 Pearson, Inc. 1.5 Parametric Relations and Inverses.

Chapter 1 Functions & Graphs Mr. J. Focht PreCalculus OHHS.

Today in Pre-Calculus Go over homework questions Notes: Inverse functions Homework.

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

Goal: Find and use inverses of linear and nonlinear functions.

Inverse Functions Given 2 functions, f(x) & g(x), if f(g(x))=x AND g(f(x))=x, then f(x) & g(x) are inverses of each other. Symbols: f -1(x) means “f.

 Another natural way to define relations is to define both elements of the ordered pair (x, y), in terms of another variable t, called a parameter 

3.4 Use Inverse Functions p. 190 What is an inverse relation?

Functions: f(x)= means y= A relation is a function if every x value is paired with one y value (no repeat x’s) A relation is a function if it passes the.

Warm Ups! Find f(g(x)) and g(f(x)) for each of the following: 1.F(x)= 2x +1, g(x) = (x-1)/2 2.F(x) = ½ x + 3, g(x) = 2x-6.

are said to be if and only if and At the same time.

Review Relation – a mapping of input values (x-values) onto output values (y-values). Here are 3 ways to show the same relation. y = x 2 x y

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.

Warm Up Solve each equation for y. 1.x = -4y 2.x = 2y x = (y + 3)/3 4.x = -1/3 (y + 1)

Section 2.6 Inverse Functions. Definition: Inverse The inverse of an invertible function f is the function f (read “f inverse”) where the ordered pairs.

Chapter 2 Functions and Graphs Copyright © 2014, 2010, 2007 Pearson Education, Inc Inverse Functions.

7.4 Inverse Functions p. 422 What is an inverse relation? What do you switch to find an inverse relation? What notation is used for an inverse function?

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

Inverse Functions. Definition A function is a set of ordered pairs with no two first elements alike. f(x) = { (x,y) : (3, 2), (1, 4), (7, 6), (9,12) }

Graphing Rational Functions Day 3. Graph with 2 Vertical Asymptotes Step 1Factor:

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.

5.3 INVERSE FUNCTIONS OBJECTIVES VERIFY ONE FUNCTION IS THE INVERSE OF ANOTHER DETERMINE WHETHER A FUNCTION HAS AN INVERSE FIND THE DERIVATIVE OF AN INVERSE.

Objective I can solve systems of equations using elimination with addition and subtraction.

Objectives: To find inverse functions graphically & algebraically.

Warm Up Solve for x in terms of y

Building Functions From Functions

Watch this!! The Inverse Function Watch this!!

Prepare for a quiz!! Pencils, please!! No Talking!!

Module 6 Review Inverses Table of Contents

Inverse Functions 5.3 Chapter 5 Functions 5.3.1

Warm-up: Given f(x) = 2x3 + 5 and g(x) = x2 – 3 Find (f ° g)(x)

Inverse Relations and Functions

Section 1.5 Inverse Functions

Precalculus Chapter 1 Section 5

Inverse Functions.

1.9 Inverse Functions f-1(x) Inverse functions have symmetry

Solving Systems of Equations

Functions and Their Inverses

Functions and Their Inverses

Composition of Functions And Inverse Functions.

4-5 Inverse Functions.

6.4 Use Inverse Functions.

3 Inverse Functions.

Sec. 2.7 Inverse Functions.

Section 7.4 Inverse Functions

Drill 1) What quadrant would each point be located in:

7.4 Inverse Functions p. 422.

Section 4.1: Inverses If the functions f and g satisfy two conditions:

7.4 Inverse Functions.

Functions and Their Inverses

The student will be able to:

Presentation transcript:

Parametric and Inverse Functions Nate Hutnik Aidan Lindvig Craig Freeh

Intro Video

Defining a Function Parametrically Parameter: Another way to define functions by defining the X and Y in terms of variable T. (x = t + 1) (y = t 2 + 2t) Now say that t=2, then just plug the number 2 into each equation to get the x and y coordinates. They are (3,8) The relationship can also be solved with substitution. Solve for t in terms of x. t = x - 1. Then plug that into the y equation. Then you end up with y = x 2 - 1, which is the same as just solving for each point with t. Tips: a graphing calculator can be used to graph it parametrically, it gives an x= and a y= to enter in the relationship of t.

Inverse relations and functions Inverse relation: The ordered pair (a,b) is in a relation if and only if the ordered pair (b,a) is in it. Horizontal line test: We've heard of the vertical line test, so the horizontal line test simply test the first equation, so see if the inverse will be a function or not. The intial graph failed the horizontal line test, so the inverse that opens to the right failed the vertical test. Inverse is f -1

Finding Inverse Algebraically Simply switch the x's and y's, and then solve it again for y. Note the domain and range. x = y/y+1 Example x(y+1) = y multiply by y+1 xy + x = y distribute xy - y = -x isolate y y(x-1) = -x factor out y y = -x/x-1 divide by x-1 y = x/1-x multiply top and bottom by -1

Inverse with reflections The points (a,b) and (b,a) are reflections of across the line y=x. they both have inverses because they passed the horizonal line test.

Verifying Inverse Functions Inverse rule: A function is an inverse if f(g(x) = x and g(f(x) = x. Example- f(x)=3x-2 g(x)=x+2/3 f(g(x) = 3(x+2/3)-2 = x+2-2 = x g(f(x) = 3x-2+2/3 = 3x/3 = x

Examples Find the value of the parameter: x = 4t and y = t for t=2 x = 4(2) x = 8 y = (2) y = y = 8 (8,9) Points can each be found like this..... or Take previous two parameters and make an equation of y in terms of x. t=x/4 y=(x/4) y = (x 2 /16) +5...either way works, and the same relationship is there.

Examples (a) is it a function (b) is its inverse a function (a) yes, because it passes the vertical line test. (b) no, because it failes the horizontal line test.