1 Example 3 (a) Find the range of f(x) = 1/x with domain [1,  ). Solution The function f is decreasing on the interval [1,  ) from its largest value.

Slides:

Advertisements

Similar presentations

Increasing/Decreasing

Advertisements

1 Example 7 Sketch the graph of the function Solution I. Intercepts The x-intercepts occur when the numerator of r(x) is zero i.e. when x=0. The y-intercept.

Math 143 Final Review Spring 2007

Transformations Transforming Graphs. 7/9/2013 Transformations of Graphs 2 Basic Transformations Restructuring Graphs Vertical Translation f(x) to f(x)

1 Example 1 (a) Let f be the rule which assigns each number to its square. Solution The rule f is given by the formula f(x) = x 2 for all numbers x. Hence.

1 Example 1 Explain why the Intermediate Value Theorem does or does not apply to each of the following functions. (a) f(x) = 1/x with domain [1,2]. Solution.

Logarithmic Functions Section 2. Objectives Change Exponential Expressions to Logarithmic Expressions and Logarithmic Expressions to Exponential Expressions.

Rational Functions Find the Domain of a function

Section 3.2 Notes Writing the equation of a function given the transformations to a parent function.

1 Example 2 Sketch the graph of the function Solution Observe that g is an even function, and hence its graph is symmetric with respect to the y-axis.

Transformations xf(x) Domain: Range:. Transformations Vertical Shifts (or Slides) moves the graph of f(x) up k units. (add k to all of the y-values) moves.

f /(x) = 6x2 – 6x – 12 = 6(x2 – x – 2) = 6(x-2)(x+1).

1 The graphs of many functions are transformations of the graphs of very basic functions. The graph of y = –x 2 is the reflection of the graph of y = x.

LIMITS INVOLVING INFINITY Mrs. Erickson Limits Involving Infinity Definition: y = b is a horizontal asymptote if either lim f(x) = b or lim f(x) = b.

Section 6.3 – Exponential Functions Laws of Exponents If s, t, a, and b are real numbers where a > 0 and b > 0, then: Definition: “a” is a positive real.

Homework: p , 17-25, 45-47, 67-73, all odd!

Lesson 2.6 Read: Pages Page 152: #1-37 (EOO), 47, 49, 51.

Limits Involving Infinity North Dakota Sunset. As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal asymptote.

SECTION 2.2 Absolute Value Functions. A BSOLUTE V ALUE There are a few ways to describe what is meant by the absolute value |x| of a real number x You.

Domains & Ranges I LOVE Parametric Equations Operations of Functions Inverse Functions Difference.

Chapter 7 Polynomial and Rational Functions with Applications Section 7.2.

Exponential Functions MM3A2e Investigate characteristics: domain and range, asymptotes, zeros, intercepts, intervals of increase and decrease, rate of.

Sullivan Algebra and Trigonometry: Section 4.4 Rational Functions II: Analyzing Graphs Objectives Analyze the Graph of a Rational Function.

What is the symmetry? f(x)= x 3 –x.

Exponential Functions and Their Graphs 2 The exponential function f with base a is defined by f(x) = a x where a > 0, a  1, and x is any real number.

Section 1.2 Analyzing Graphs x is the distance from the y-axis f(x) is the distance from the x-axis p. 79 Figure 1.6 (open dot means the graph does not.

Translations and Combinations Algebra 5/Trigonometry.

Transformations of Functions. Graphs of Common Functions See Table 1.4, pg 184. Characteristics of Functions: 1.Domain 2.Range 3.Intervals where its increasing,

Review Limits When you see the words… This is what you think of doing…  f is continuous at x = a  Test each of the following 1.

1 Example 2 (a) Find the range of f(x) = x 2 +1 with domain [0,2]. Solution The function f is increasing on the interval [0,2] from its smallest value.

2-6 rational functions.  Lines l and m are perpendicular lines that intersect at the origin. If line l passes through the point (2,-1), then line m must.

3.4 Properties of Logarithmic Functions

5.4 Logarithmic Functions. Quiz What’s the domain of f(x) = log x?

1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 3 Polynomial and Rational Functions.

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

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 4.3, Slide 1 Chapter 4 Exponential Functions.

L AST MATHEMATICIAN STANDING Review P ROBLEM 1 Find f(g(x)) and f(f(x)).

Exponential & Logarithmic functions. Exponential Functions y= a x ; 1 ≠ a > 0,that’s a is a positive fraction or a number greater than 1 Case(1): a >

2.5 Shifting, Reflecting, and Stretching Graphs. Shifting Graphs Digital Lesson.

(a) (b) (c) (d) Warm Up: Show YOUR work!. Warm Up.

Rational Functions Objective: Finding the domain of a rational function and finding asymptotes.

Section 1.4 Transformations and Operations on Functions.

3.1 Exponential and Logistic Functions. Exponential functions Let a and b real number constants. An exponential function in x is a function that can be.

Math 140 Quiz 3 - Summer 2004 Solution Review (Small white numbers next to problem number represent its difficulty as per cent getting it wrong.)

Exponential & Logarithmic functions. Exponential Functions y= a x ; 1 ≠ a > 0,that’s a is a positive fraction or a number greater than 1 Case(1): a >

The Twelve Basic Functions Section 1.3 Pgs 102 – 103 are very important!

2.6. A rational function is of the form f(x) = where N(x) and D(x) are polynomials and D(x) is NOT the zero polynomial. The domain of the rational function.

Twenty Questions Rational Functions Twenty Questions

Transformations of Functions. The vertex of the parabola is at (h, k).

3.1 Exponential Functions and Their Graphs Objectives: Students will recognize and evaluate exponential functions with base a. Students will graph exponential.

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

REVIEW. A. All real numbers B. All real numbers, x ≠ -5 and x ≠ -2 C. All real numbers, x ≠ 2 D. All real numbers, x ≠ 5 and x ≠ 2.

1.5 Analyzing Graphs of Functions (-1,-5) (2,4) (4,0) Find: a.the domain b.the range c.f(-1) = d.f(2) = [-1,4) [-5,4] -5 4.

26 – Limits and Continuity II – Day 2 No Calculator

2.2 Limits Involving Infinity

Precalculus Day 46.

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

Rational Functions, Transformations

Partner Whiteboard Review.

Part (a) h(1) = f(g(1)) - 6 h(3) = f(g(3)) - 6 h(1) = f(2) - 6

1.4 Continuity and One-Sided Limits

In symbol, we write this as

In symbol, we write this as

Pre-Calculus Go over homework End behavior of a graph

Ex1 Which mapping represents a function? Explain

27 – Graphing Rational Functions No Calculator

Exponential Functions and Their Graphs

One-to-One Functions;

Exponential Functions and Their Graphs

f(x) g(x) x x (-8,5) (8,4) (8,3) (3,0) (-4,-1) (-7,-1) (3,-2) (0,-3)

Presentation transcript:

1 Example 3 (a) Find the range of f(x) = 1/x with domain [1,  ). Solution The function f is decreasing on the interval [1,  ) from its largest value f(1) = 1 towards its horizontal asymptote y = 0. Hence the range of f is contained in the interval (0,1]. Since the function f is continuous on the interval [1,  ), the Intermediate Value Theorem applies to f. Hence the range of f is an interval from a to 1. Since the values of f are all positive, a  0. Since the x-axis is a horizontal asymptote of f, there are large numbers x for which f(x)  a are arbitrarily close to zero. Hence a = 0, and the range of f is the interval (0,1].

2 (b) Find the range of g(x) = 1/x with domain (0,1]. Solution The function g is decreasing on the interval (0,1] from its vertical asymptote x = 0 towards its smallest value g(1) = 1. Hence the range of g is contained in the interval [1,  ). Since the function g is continuous on the interval (0,1], the Intermediate Value Theorem applies to g. Hence the range of g is an interval from 1 to b. Since the y-axis is a vertical asymptote of g there are numbers x close to 0 for which g(x)  b are arbitrarily large. Hence b = , and the range of g is the interval [1,  ).

3 (c) Find the range of h(x) = 1/x with domain (0,  ). Solution The function h is decreasing on the interval (0,  ) from its vertical asymptote x = 0 towards its horizontal asymptote y = 0. Since the values of h are all positive, the range of h is contained in the interval (0,  ). The function h is continuous on the interval (0,  ), the Intermediate Value Theorem applies to h. Hence the range of h is an interval from a to b. Since the y-axis is a vertical asymptote of h there are numbers x close to 0 for which h(x)  b are arbitrarily large. Hence b = . Since the x-axis is a horizontal asymptote of h there are large numbers x for which h(x)  a are arbitrarily close to zero. Hence a=0 and the range of h is the interval (0,  ).