Graphs of Functions and Inverses What is the connection between the graphs of functions and their inverses?

Slides:

Advertisements

Similar presentations

Warm Up 1. 5x – 2 when x = – t 2 when 3. when x = Give the domain and range for this relation: {(1, 1), (–1, 1), (2, 4), (–2, 4),

Advertisements

Composite Functions f(x)=x 2 g(x)=x+5h(x)= 1 / x Find as many composite functions made using two of these as you can.

MAPPINGS and FUNCTIONS

Math – Getting Information from the Graph of a Function 1.

Finding the Inverse. 1 st example, begin with your function f(x) = 3x – 7 replace f(x) with y y = 3x - 7 Interchange x and y to find the inverse x = 3y.

Functions A function is a relationship between two sets: the domain (input) and the range (output) DomainRange Input Output This.

Objectives Write equivalent forms for exponential and logarithmic functions. Write, evaluate, and graph logarithmic functions.

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.

Holt Algebra Logarithmic Functions Write equivalent forms for exponential and logarithmic functions. Write, evaluate, and graph logarithmic functions.

Job 1 Pays £10 per day. Total Earnings £300 Pays 1p per day but daily pay increases by 50% each day. Total Earnings £ Summer Jobs - 30 days work.

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,

Holt Algebra Identifying Quadratic Functions 9-1 Identifying Quadratic Functions Holt Algebra 1 Warm Up Warm Up Lesson Presentation Lesson Presentation.

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) =

Lesson 28 – Radical Functions Math 2 Honors - Santowski.

Unit 1B quadratics Day 3. Graphing a Quadratic Function EQ: How do we graph a quadratic function that is in vertex form? M2 Unit 1B: Day 3 Lesson 3.1B.

IGCSE FM Domain/Range Dr J Frost Last modified: 14 th October 2015 Objectives: The specification:

Math – What is a Function? 1. 2 input output function.

Harder one: A function is defined by : f(x) = (x+3)(x-5) Find the inverse and state the domain and range of f(x) and f -1 (x)

Functions and Their Inverses

Inverses of Relations and Functions

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

Starter Easy (hopefully) sub values into functions card activity.

Calculating Algebraic Inverses. Daily Check Find the inverse of the following functions.

2.2 Square Root of a Function Square Root of the Linear Function

Holt McDougal Algebra Logarithmic Functions Write equivalent forms for exponential and logarithmic functions. Write, evaluate, and graph logarithmic.

Write a function rule for a graph EXAMPLE 3 Write a rule for the function represented by the graph. Identify the domain and the range of the function.

Lesson 3.6 (Continued) Graphing Exponential Functions : Graphing Exponential Functions.

Aims: To be able to find the inverse of a function. To know the graphical relationship between a function and its inverse. To understand the relationship.

Objective: Students will be able to write equivalent forms for exponential and logarithmic functions, and can write, evaluate, and graph logarithmic functions.

Unit 1: Functions Minds On. Unit 1: Functions Lesson 5: Inverse Functions Example: Find the inverse of f(x) = (x – 3)

Inverse functions  Recap of inverse of a function.  Inverse functions with e x and ln x.

Algebra 2 Inverse Relations and Functions Lesson 7-7.

6/12/2016HL1 Math - Santowski1 Lesson 10 – Inverses & Inverse Functions IBHL1 Math - Santowski 6/12/20161.

Inverse Relations and Functions

GCSE/IGCSE-FM Functions

Quadratic and Square Root Inverse Relationships with Restrictions

Functions 12-4 Warm Up Problem of the Day Lesson Presentation

Graphing Relationships

CIE Centre A-level Pure Maths

a. Write a function, f(x), to represent the 5 bonus points.

Inverse Functions Algebra III, Sec. 1.9 Objective

Lesson 1 -Introduction to Functions

Warmup Let f(x) = x – 3 and g(x) = x2. What is (f ○ g)(1)?

INVERSE FUNCTIONS.

Notes Over 2.1 Function {- 3, - 1, 1, 2 } { 0, 2, 5 }

Functions Review.

2.2 Square Root of a Function Square Root of the Linear Function

Objective 1A f(x) = 2x + 3 What is the Range of the function

Identifying Quadratic Functions

Before: October 19, 2017 A function is represented by f (x) = 2x + 5. Find f (3). Create a table representing the above function with a domain: {1,2,3,4}

Lesson 30 – Radical Functions

INVERSE FUNCTIONS.

Lesson 5.8 Graphing Polynomials.

3.3 The Inverse of a Quadratic Function

Quadratics Lesson 2 Objective: Vertex Form of a Quadratic.

Functions Inverses.

Functions Inverses.

Math log #29 1/30/19 a.

Warm Up #8 Sketch the graphs of: 1.

Alegebra 2A Function Lesson 1 Objective: Relations, and Functions.

Composite Function: Combining a function within another function.

Graphing Quadratic Functions

Inverse Functions A function and its inverse function can be described as the "DO" and the "UNDO" functions. A function takes a starting value, performs.

More Functions L.O. All pupils can recall the different types of functions All pupils can find the domain and range of basic functions Some pupils can.

2.1(b) Notes: Graphing Quadratic Functions

Equations & Graphing Algebra 1, Unit 3, Lesson 5.

Presentation transcript:

Graphs of Functions and Inverses What is the connection between the graphs of functions and their inverses?

Inverse Functions Aims: To find harder inverse functions for quadratics and algebraic fractions.

Learning Outcomes Name: What is an inverse function? (continued from last lesson) Describe: How to limit a quadratic function so that it is one-one. How to find an inverse function for quadratic functions. Explain: The connection between the domain and range of functions and their inverses.

Continuing Our Work This lesson follows on from last lesson and deals with the types of problem that may be seen in Core 3 exams. The problems will involve… – finding domains and ranges of functions –finding inverse functions –using the relationship between the domains and ranges of functions and their inverses.

Example 1 Algebraic Fractions Work out f -1 (x)? What are the domain and range of f -1 (x)? Sketch f -1 (x)

Example 2 Quadratics Complete the square to find the minimum point; use this to write the range of f(x). Use the completed square form to find f -1 (x) Sketch the graphs of f(x) and f -1 (x)

Example 2 Quadratics Complete the square to find the minimum point; use this to write the range of f(x). Use the completed square form to find f -1 (x) Sketch the graphs of f(x) and f -1 (x)

The Story So Far... (might seem familiar) The evil maths wizard Moldywart has been up to his old tricks again in an effort to remove his blemishes and generally do nasty stuff! It is up to Hairry, Rong and Hurtmyknee to reverse his wicked spells!... and it is your job to help them!

Your Task To reverse his spells (functions) you must have all the right ingredients… –You must have the reverse spell (inverse function) –You must know where the inverse spell must be cast from and to (its domain and range) –The path that the spell must take (a sketch of its graph)

Learning Outcomes Name: What is an inverse function? (continued from last lesson) Describe: How to limit a quadratic function so that it is one-one. How to find an inverse function for quadratic functions. Explain: The connection between the domain and range of functions and their inverses.

Learning Outcomes Name: An inverse function is a function that reverses a function (e.g. it turns each output back into the input that made it) Describe: To inverse a quadratic you must check that the domain means that it is a one-one function (this can be found from the completed square form) x≤or≥ minimum point x value. The completed square form is then made equal to y and the function is rearranged to find x in terms of y. The function in y is the inverse function. Explain: The domain of the function becomes the range of the inverse function and vice versa. Since domains are easier to identify than ranges this can be an effective way to identify the range of a function.