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## Module 1: Functions and Graphs

More function types, learning outcomes.

• Identify a rational function
• Describe the graphs of power and root functions
• Explain the difference between algebraic and transcendental functions
• Graph a piecewise-defined function

## Algebraic Functions

By allowing for quotients and fractional powers in polynomial functions, we create a larger class of functions. An algebraic function is one that involves addition, subtraction, multiplication, division, rational powers, and roots. Two types of algebraic functions are rational functions and root functions.

Just as rational numbers are quotients of integers, rational functions are quotients of polynomials. In particular, a rational function is any function of the form $f(x)=p(x)/q(x)$, where $p(x)$ and $q(x)$ are polynomials. For example,

are rational functions. A root function is a power function of the form $f(x)=x^{1/n}$, where $n$ is a positive integer greater than one. For example, $f(x)=x^{1/2}=\sqrt{x}$ is the square-root function and $g(x)=x^{1/3}=\sqrt[3]{x}$ is the cube-root function. By allowing for compositions of root functions and rational functions, we can create other algebraic functions. For example, $f(x)=\sqrt{4-x^2}$ is an algebraic function.

## Example: Finding Domain and Range for Algebraic Functions

For each of the following functions, find the domain and range.

• $f(x)=\dfrac{3x-1}{5x+2}$
• $f(x)=\sqrt{4-x^2}$

When we multiply both sides of this equation by $5x+2$, we see that $x$ must satisfy the equation

From this equation, we can see that $x$ must satisfy

If $y=\frac{3}{5}$, this equation has no solution. On the other hand, as long as $y\ne \frac{3}{5}$,

These two inequalities reduce to $2 \ge x$ and $x \ge -2$. Therefore, the set $\{x|-2\le x\le 2\}$ must be part of the domain. For both terms to be negative, we need

If $-2 \le x \le 2$, then $0 \le 4-x^2 \le 4$. Therefore, $0 \le \sqrt{4-x^2} \le 2$, and the range of $f$ is $\{y|0 \le y \le 2\}$.

Find the domain and range for the function $f(x)=\dfrac{(5x+2)}{(2x-1)}$.

The denominator cannot be zero. Solve the equation $y=\frac{(5x+2)}{(2x-1)}$ for $x$ to find the range.

The domain is the set of real numbers $x$ such that $x \ne \frac{1}{2}$. The range is the set $\{y|y \ne \frac{5}{2}\}$.

The root functions $f(x)=x^{1/n}$ have defining characteristics depending on whether $n$ is odd or even. For all even integers $n \ge 2$, the domain of $f(x)=x^{1/n}$ is the interval $[0,\infty)$. For all odd integers $n \ge 1$, the domain of $f(x)=x^{1/n}$ is the set of all real numbers. Since $x^{1/n}=(−x)^{1/n}$ for odd integers $n, \, f(x)=x^{1/n}$ is an odd function if $n$ is odd. See the graphs of root functions for different values of $n$ in Figure 11.

Figure 11. (a) If $n$ is even, the domain of $f(x)=\sqrt[n]{x}$ is $[0,\infty)$. (b) If $n$ is odd, the domain of $f(x)=\sqrt[n]{x}$ is $(-\infty,\infty )$ and the function $f(x)=\sqrt[n]{x}$ is an odd function.

## Example: Finding Domains for Algebraic Functions

For each of the following functions, determine the domain of the function.

• $f(x)=\dfrac{3}{x^2-1}$
• $f(x)=\dfrac{2x+5}{3x^2+4}$
• $f(x)=\sqrt{4-3x}$
• $f(x)=\sqrt[3]{2x-1}$
• You cannot divide by zero, so the domain is the set of values $x$ such that $x^2-1 \ne 0$. Therefore, the domain is $\{x|x \ne \pm 1\}$.
• You need to determine the values of $x$ for which the denominator is zero. Since $3x^2+4 \ge 4$ for all real numbers $x$, the denominator is never zero. Therefore, the domain is $(-\infty,\infty )$.
• Since the square root of a negative number is not a real number, the domain is the set of values $x$ for which $4-3x \ge 0$. Therefore, the domain is $\{x|x \le \frac{4}{3}\}$.
• The cube root is defined for all real numbers, so the domain is the interval $(-\infty, \infty)$.

Find the domain for each of the following functions: $f(x)=\dfrac{(5-2x)}{(x^2+2)}$ and $g(x)=\sqrt{5x-1}$.

Determine the values of $x$ when the expression in the denominator of $f$ is nonzero, and find the values of $x$ when the expression inside the radical of $g$ is nonnegative.

The domain of $f$ is $(-\infty, \infty)$ The domain of $g$ is $\{x|x \ge \frac{1}{5}\}$.

## Transcendental Functions

Thus far, we have discussed algebraic functions. Some functions, however, cannot be described by basic algebraic operations. These functions are known as transcendental functions because they are said to “transcend,” or go beyond, algebra. The most common transcendental functions are trigonometric, exponential, and logarithmic functions. A trigonometric function relates the ratios of two sides of a right triangle. They are $\sin x,\, \cos x, \, \tan x, \, \cot x,\, \sec x$, and $\csc x$. (We discuss trigonometric functions later in the module.) An exponential function is a function of the form $f(x)=b^x$, where the base $b>0, \, b \ne 1$. A logarithmic function is a function of the form $f(x)=\log_b(x)$ for some constant $b>0, \, b \ne 1$, where $\log_b(x)=y$ if and only if $b^y=x$. (We also discuss exponential and logarithmic functions later in the module.)

## Example: Classifying Algebraic and Transcendental Functions

Classify each of the following functions, a. through c., as algebraic or transcendental.

• $f(x)= \dfrac{\sqrt{x^3+1}}{4x+2}$
• $f(x)=2^{x^2}$
• $f(x)=\sin (2x)$
• Since this function involves basic algebraic operations only, it is an algebraic function.
• This function cannot be written as a formula that involves only basic algebraic operations, so it is transcendental. (Note that algebraic functions can only have powers that are rational numbers.)
• As in part (b), this function cannot be written using a formula involving basic algebraic operations only; therefore, this function is transcendental.

Watch the following video to see the worked solution to Example: Classifying Algebraic and Transcendental Functions

Is $f(x)=\dfrac{x}{2}$ an algebraic or a transcendental function?

## Piecewise-Defined Functions

Sometimes a function is defined by different formulas on different parts of its domain. A function with this property is known as a piecewise-defined function . The absolute value function is an example of a piecewise-defined function because the formula changes with the sign of $x$:

Other piecewise-defined functions may be represented by completely different formulas, depending on the part of the domain in which a point falls. To graph a piecewise-defined function, we graph each part of the function in its respective domain, on the same coordinate system. If the formula for a function is different for $x<a$ and $x>a$, we need to pay special attention to what happens at $x=a$ when we graph the function. Sometimes the graph needs to include an open or closed circle to indicate the value of the function at $x=a$. We examine this in the next example. If you need a refresher, check out the Recall box first.

## Recall: Given a piecewise function, sketch a graph.

• Indicate on the $x$-axis the boundaries defined by the intervals on each piece of the domain.
• For each piece of the domain, graph on that interval using the corresponding equation pertaining to that piece. Do not graph two functions over one interval because it would violate the criteria of a function.

## Example: Graphing a Piecewise-Defined Function

Sketch a graph of the following piecewise-defined function:

$f(x)=\begin{cases} x+3, & x < 1 \\ (x-2)^2 & x \ge 1 \end{cases}$

Graph the linear function $y=x+3$ on the interval $(-\infty,1)$ and graph the quadratic function $y=(x-2)^2$ on the interval $[1,\infty )$. Since the value of the function at $x=1$ is given by the formula $f(x)=(x-2)^2$, we see that $f(1)=1$. To indicate this on the graph, we draw a closed circle at the point $(1,1)$. The value of the function is given by $f(x)=x+2$ for all $x<1$, but not at $x=1$. To indicate this on the graph, we draw an open circle at $(1,4)$.

Figure 12. This piecewise-defined function is linear for $x<1$ and quadratic for $x \ge 1$.

Sketch a graph of the function

$f(x)=\begin{cases} 2-x, & x \le 2 \\ x+2, & x>2 \end{cases}$

Graph one linear function for $x \le 2$ and then graph a different linear function for $x>2$.

Figure 13. Graph of piecewise function.

## Example: Parking Fees Described by a Piecewise-Defined Function

In a big city, drivers are charged variable rates for parking in a parking garage. They are charged $10 for the first hour or any part of the first hour and an additional$2 for each hour or part thereof up to a maximum of $30 for the day. The parking garage is open from 6 a.m. to 12 midnight. • Write a piecewise-defined function that describes the cost $C$ to park in the parking garage as a function of hours parked $x$. • Sketch a graph of this function $C(x)$. • Since the parking garage is open 18 hours each day, the domain for this function is $\{x|0 < x \le 18\}$. The cost to park a car at this parking garage can be described piecewise by the function $C(x)=\begin{cases} \\ 10, & 0 < x \le 1 \\ 12, & 1 < x \le 2 \\ 14, & 2 < x \le 3 \\ 16, & 3 < x \le 4 \\ & \vdots \\ 30, & 10 < x \le 18 \end{cases}$ Figure 14. Graph of parking fees vs. hours spent parked in garage. Watch the following video to see the worked solution to Example: Parking Fees Described by a Piecewise-Defined Function The cost of mailing a letter is a function of the weight of the letter. Suppose the cost of mailing a letter is $49\text{¢}$ for the first ounce and $21\text{¢}$ for each additional ounce. Write a piecewise-defined function describing the cost $C$ as a function of the weight $x$ for $0 < x \le 3$, where $C$ is measured in cents and $x$ is measured in ounces. The piecewise-defined function is constant on the intervals $(0,1], \, (1,2], \, \cdots$ $C(x)=\begin{cases} 49, & 0 < x \le 1 \\ 70, & 1 < x \le 2 \\ 91, & 2 < x \le 3 \end{cases}$ • 1.2 Basic Classes of Functions. Authored by : Ryan Melton. License : CC BY: Attribution • Calculus Volume 1. Authored by : Gilbert Strang, Edwin (Jed) Herman. Provided by : OpenStax. Located at : https://openstax.org/details/books/calculus-volume-1 . License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike . License Terms : Access for free at https://openstax.org/books/calculus-volume-1/pages/1-introduction Privacy Policy If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. ## Integrated math 1 Course: integrated math 1 > unit 8. • Unit test Functions #### IMAGES 1. [Solved] Math 2. 1.10 Part 2 Test.docx 3. 7.09 Graded Assignment Sequences and Modeling with Functions 4. SOLUTION: Graded Assignment Unit Test Part 2 More Function Types 5. [Solved] Graded Assignment Unit Test, Part 2 Complete this teacher 6. Polynomials Unit Test Part 2.pdf #### VIDEO 1. School Topper 2. SAT Math: Evaluating Functions 3. Week 1 Graded Assignment Solution 4. Boost Unit test Part 2 5. Class 9 Life Science || 3rd Unit Test Suggestions || Part 1 || Chapter 1 & Chapter 2 6. 1.Cell- basic unit of life #### COMMENTS 1. Unit 2: More Function Types Flashcards a second-degree polynomial whose graph is a parabola. parabola. the U-shaped graph of a quadratic function. x-intercept. a point where a function touches or crosses the x-axis. axis of symmetry. the vertical line that separates the graph of a quadratic function into two halves that are reflections of each other. 2. Solved Graded AssignmentUnit Test, Part 2: More Function Algebra questions and answers. Graded AssignmentUnit Test, Part 2: More Function TypesAnswer the questions below. When you are finished, submit this test to your teacher for full credit.Total score: , ﻿of 15 ﻿points (Score for Question 1: q, ﻿of 5 ﻿points)Graph the function f (x)=x-23.Answer:f (x) (C) 2016 ﻿K12 ﻿Inc. 3. Unit 2: More Function Types Flashcards a family of functions of the form f (x) = C / 1 + Ae^-Bx, where A, B, and C are positive numbers. parent function. the basic function in a family of functions. piecewise function. a function that is defined using different rules for different intervals of the domain. Unit 2: More Function Types. 4. Graded Assignment Unit Test Part 2 More Function Types Questions G r a d ed As si gn m en t. Unit Test, Part 2: More Function Types. Answer the questions b elow. When yo u are finished, s ubmit this test to your te acher for full credit. Instructions: Please don't use a g raphing calculat or. All graphs need to be drawn by hand. Total score: ____ of 1 6 points. 5. PDF Unit Test, Part 2: More Function Types This assignment must be printed, completed by hand, and scanned to the Assignments dropbox for credit. This will be graded by your teacher. Show all work. 1. Graph the function = 6 −| − 4 3 5 |. 2. Use what you learned in unit 2 to describe the function. What is the parent function and what general shape does the parent function have ... 6. Math Question: Math | ﻿Graded Assignment | ﻿Unit Test, Part 2 | ﻿More Function Types(Score for Question 3: q, ﻿of 5 ﻿points)3. ﻿Graph the function f(x)=-(x-2)2+4.Answer: Math | ﻿Graded Assignment | ﻿Unit Test, Part 2 | ﻿More Function 7. algebra 2b Study with Quizlet and memorize flashcards containing terms like lesson 21, compare the function f(x) = x^2 − 3x − 10 with the following graph of g(x). which statements about the intercepts are true? select all that apply., compare the function f(x) = √x+2 − 1 with the following graph of g(x). which statement about the x-intercepts for f(x) and g(x) is true? and more. 8. More Function Types By allowing for quotients and fractional powers in polynomial functions, we create a larger class of functions. An algebraic function is one that involves addition, subtraction, multiplication, division, rational powers, and roots. Two types of algebraic functions are rational functions and root functions. Just as rational numbers are quotients ... 9. Functions (part 2): Unit test Unit test. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. 10. Functions: Unit test Unit test. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. 11. 2.13 Unit 2 Test Part 2.docx Name: Date: Graded Assignment Adapted Unit Test, Part 2: More Function Types Answer the questions below. When you. AI Homework Help. Expert Help. Study Resources. ... Name: Date: Graded Assignment Adapted Unit Test, Part 2: More Function Types Answer the questions below. When you are finished, submit this test to your teacher for full credit ... 12. 10.12 Graded Assignment More Function Types View 10.12 Graded Assignment_ More Function Types - Part 2.docx from MATH Algebra 2 at Keystone National High School. Graded Assignment: More Function Types - Part 2 Skyla Jones Ms. ... 6 10 Suppose that the following Dickey Fuller test regression is conducted and. document. Module 18_ Transformations Around the Globe.docx. NetTech_Exam(DS2)(Q1 ... 13. 11.08 Graded Assignment Using Function Models Graded Assignment Unit Test, Part 2: Sequences and Series Answer the questions below. When you are finished, submit this test to your teacher for full credit. Total score: ____ of 15 points (Score ... 10.12 Graded Assignment_ More Function Types - Part 2.docx. Keystone National High School. MATH Algebra 2. Skyla Jones. Ms McGraw. 14. Solved Graded Assignment Unit Test, Part 2: Quadratic Graded Assignment Unit Test, Part 2: Quadratic Equations Answer the questions below. When you are finished, submit this test to your teacher for full credit. Total score: of 15 points (Score for Question 1: _ of 6 points) 1. Explain why the equation (x − 4) 2 − 28 = 8 has two solutions. Then solve the equation to find the solutions Show ... 15. Algebra 2.13: Unit Test Flashcards x−4.76=−7, Solve for x. 2/3x=10 and more. ... K12, 2.12 Unit Test Algebra: One-Variable Linear Equations and Inequalities - Part 1 ... Part 1. 12 terms. kayla_jaimes4. Preview. Lecture 2 Canine and Feline Breeds and Life Stages. 89 terms. FrancesWBryant. Preview. Algebra II - Chapter 1: Linear Functions and Systems. 17 terms. 16. PDF Unit Test, Part 2: Quadratic Equations Graded Assignment Unit Test, Part 2: Quadratic Equations Answer the questions below. When you are finished, submit this test to your teacher for full credit. 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Graded Assignment Unit Test, Part 2: Quadratic Functions Answer the questions below. When you are finished, submit this test to your teacher for full credit. Total score: ____ of 15 points (Score for Question 1: ___ of 5 points) 1. Graph the function f x x x 2 4 12 on the coordinate plane. (a) What are the x-intercepts? (b) What is the y-intercept?

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Answered step-by-step. Asked by skbboys123. Graded Assignment Unit Test, Part 2: Sequences and Series Answer the questions below. When you are finished, submit this test to your teacher for full credit. Total score: ____ of 15 points (Score for Question 1: ___ of 5 points) 1. Find the sum of the series 15 1 (2 1) n n .