Can a Graph Have Two Horizontal Asymptotes Although most

In Exercises 1-4, solve the equation or inequality. x2 - 16 = 0

In Exercises 1-4, solve the equation or inequality. 9-x2=0

In Exercises 1-4, solve the equation or inequality. x - 10 < 0

In Exercises 1-4, solve the equation or inequality. 5 - x \(\leq\) 0

In Exercises 5-10, find all values of x algebraically for which the algebraic expression is not defined. Support your answer graphically. \(\frac{x}{x-16}\)

In Exercises 5-10, find all values of x algebraically for which the algebraic expression is not defined. Support your answer graphically. \(\frac{x}{x^{2}-16}\)

In Exercises 5-10, find all values of x algebraically for which the algebraic expression is not defined. Support your answer graphically. \(\sqrt{x-16}\)

In Exercises 5-10, find all values of x algebraically for which the algebraic expression is not defined. Support your answer graphically. \(\frac{\sqrt{x^{2}+1}}{x^{2}-1}\)

In Exercises 5-10, find all values of x algebraically for which the algebraic expression is not defined. Support your answer graphically. \(\frac{\sqrt{x+2}}{\sqrt{3-x}}\)

In Exercises 5-10, find all values of x algebraically for which the algebraic expression is not defined. Support your answer graphically. \(\frac{x^{2}-2 x}{x^{2}-4}\)

In Exercises 1-4, determine whether the formula determines y as a function of x. If not, explain why not \(y=\sqrt{x-4}\)

In Exercises 1-4, determine whether the formula determines y as a function of x. If not, explain why not \(y=x^{2} \pm 3\)

In Exercises 1-4, determine whether the formula determines y as a function of x. If not, explain why not X = 2y2

In Exercises 1-4, determine whether the formula determines y as a function of x. If not, explain why not x = 12 - y

In Exercises 5-8, use the vertical line test to determine whether the curve is the graph of a function.

In Exercises 5-8, use the vertical line test to determine whether the curve is the graph of a function.

In Exercises 5-8, use the vertical line test to determine whether the curve is the graph of a function.

In Exercises 5-8, use the vertical line test to determine whether the curve is the graph of a function.

In Exercises 9-16, find the domain of the function algebraically and support your answer graphically \(f(x)=x^{2}+4\)

In Exercises 9-16, find the domain of the function algebraically and support your answer graphically \(h(x)=\frac{5}{x-3}\)

In Exercises 9-16, find the domain of the function algebraically and support your answer graphically \(f(x)=\frac{3 x-1}{(x+3)(x-1)}\)

In Exercises 9-16, find the domain of the function algebraically and support your answer graphically \(f(x)=\frac{1}{x}+\frac{5}{x-3}\)

In Exercises 9-16, find the domain of the function algebraically and support your answer graphically \(g(x)=\frac{x}{x^{2}-5 x}\)

In Exercises 9-16, find the domain of the function algebraically and support your answer graphically \(h(x)=\frac{\sqrt{4-x^{2}}}{x-3}\)

In Exercises 9-16, find the domain of the function algebraically and support your answer graphically \(h(x)=\frac{\sqrt{4-x}}{(x+1)\left(x^{2}+1\right)}\)

In Exercises 9-16, find the domain of the function algebraically and support your answer graphically \(f(x)=\sqrt{x^{4}-16 x^{2}}\)

In Exercises 17-20, find the range of the function. \(f(x)=10-x^{2}\)

In Exercises 17-20, find the range of the function. \(g(x)=5+\sqrt{4-x}\)

In Exercises 17-20, find the range of the function. \(f(x)=\frac{x^{2}}{1-x^{2}}\)

In Exercises 17-20, find the range of the function. \(g(x)=\frac{3+x^{2}}{4-x^{2}}\)

In Exercises 21-24, graph the function and tell whether or not it has a point of discontinuity at x=0. If there is a discontinuity, tell whether it is removable or nonremovable. \(g(x)=\frac{3}{x}’)

In Exercises 21-24, graph the function and tell whether or not it has a point of discontinuity at x=0. If there is a discontinuity, tell whether it is removable or nonremovable. \(h(x)=\frac{x^{3}+x}{x}\)

In Exercises 21-24, graph the function and tell whether or not it has a point of discontinuity at x=0. If there is a discontinuity, tell whether it is removable or nonremovable. \(f(x)=\frac{|x|}{x}\)

In Exercises 21-24, graph the function and tell whether or not it has a point of discontinuity at x=0. If there is a discontinuity, tell whether it is removable or nonremovable. \(g(x)=\frac{x}{x-2}\)

In Exercises 25-28, state whether each labeled point identifies a local minimum, a local maximum, or neither. Identify intervals on which the function is decreasing and increasing.

In Exercises 25-28, state whether each labeled point identifies a local minimum, a local maximum, or neither. Identify intervals on which the function is decreasing and increasing.

In Exercises 25-28, state whether each labeled point identifies a local minimum, a local maximum, or neither. Identify intervals on which the function is decreasing and increasing.

In Exercises 25-28, state whether each labeled point identifies a local minimum, a local maximum, or neither. Identify intervals on which the function is decreasing and increasing.

In Exercises 29-34, graph the function and identify intervals on which the function is increasing, decreasing, or constant \(f(x)=|x+2|-1\)

In Exercises 29-34, graph the function and identify intervals on which the function is increasing, decreasing, or constant \(f(x)=|x+1|+|x-1|-3\)

In Exercises 29-34, graph the function and identify intervals on which the function is increasing, decreasing, or constant \(g(x)=|x+2|+|x-1|-2\)

In Exercises 29-34, graph the function and identify intervals on which the function is increasing, decreasing, or constant \(h(x)=0.5(x+2)^{2}-1\)

In Exercises 29-34, graph the function and identify intervals on which the function is increasing, decreasing, or constant \(g(x)=3-(x-1)^{2}\)

In Exercises 29-34, graph the function and identify intervals on which the function is increasing, decreasing, or constant \(f(x)=x^{3}-x^{2}-2 x\)

In Exercises 35-40, determine whether the function is bounded above, bounded below, or bounded on its domain. y = 32

In Exercises 35-40, determine whether the function is bounded above, bounded below, or bounded on its domain. y = 2 - x2

In Exercises 35-40, determine whether the function is bounded above, bounded below, or bounded on its domain. y = 2x

In Exercises 35-40, determine whether the function is bounded above, bounded below, or bounded on its domain. y = 2-x

In Exercises 35-40, determine whether the function is bounded above, bounded below, or bounded on its domain. \(y=\sqrt{1-x^{2}}\)

In Exercises 35-40, determine whether the function is bounded above, bounded below, or bounded on its domain. \(y=x-x^{3}\)

In Exercises 41-46, use a grapher to find all local maxima and minima and the values of x where they occur. Give values rounded to two decimal places. \(f(x)=4-x+x^{2}\)

In Exercises 41-46, use a grapher to find all local maxima and minima and the values of x where they occur. Give values rounded to two decimal places. \(g(x)=x^{3}-4 x+1\)

In Exercises 41-46, use a grapher to find all local maxima and minima and the values of x where they occur. Give values rounded to two decimal places. \(h(x)=-x^{3}+2 x-3\)

In Exercises 41-46, use a grapher to find all local maxima and minima and the values of x where they occur. Give values rounded to two decimal places. \(f(x)=(x+3)(x-1)^{2}\)

In Exercises 41-46, use a grapher to find all local maxima and minima and the values of x where they occur. Give values rounded to two decimal places. \(h(x)=x^{2} \sqrt{x+4}\)

In Exercises 41-46, use a grapher to find all local maxima and minima and the values of x where they occur. Give values rounded to two decimal places. \(g(x)=x|2 x+5|\)

In Exercises 47-54, state whether the function is odd, even, or neither. Support graphically and confirm algebraically. \(f(x)=2 x^{4}\)

In Exercises 47-54, state whether the function is odd, even, or neither. Support graphically and confirm algebraically. \(g(x)=x^{3}\)

In Exercises 47-54, state whether the function is odd, even, or neither. Support graphically and confirm algebraically. \(f(x)=\sqrt{x^{2}+2}\)

In Exercises 47-54, state whether the function is odd, even, or neither. Support graphically and confirm algebraically. \(g(x)=\frac{3}{1+x^{2}}\)

In Exercises 47-54, state whether the function is odd, even, or neither. Support graphically and confirm algebraically. \(f(x)=-x^{2}+0.03 x+5\)

In Exercises 47-54, state whether the function is odd, even, or neither. Support graphically and confirm algebraically. \(f(x)=x^{3}+0.04 x^{2}+3\)

In Exercises 47-54, state whether the function is odd, even, or neither. Support graphically and confirm algebraically. \(g(x)=2 x^{3}-3 x\)

In Exercises 47-54, state whether the function is odd, even, or neither. Support graphically and confirm algebraically. \(h(x)=\frac{1}{x}\)

In Exercises 55-62, use a method of your choice to find all horizontal and vertical asymptotes of the function. \(f(x)=\frac{x}{x-1}\)

In Exercises 55-62, use a method of your choice to find all horizontal and vertical asymptotes of the function. \(q(x)=\frac{x-1}{x}\)

In Exercises 55-62, use a method of your choice to find all horizontal and vertical asymptotes of the function. \(g(x)=\frac{x+2}{3-x}\)

In Exercises 55-62, use a method of your choice to find all horizontal and vertical asymptotes of the function. \(q(x)=1.5^{x}\)

In Exercises 55-62, use a method of your choice to find all horizontal and vertical asymptotes of the function. \(f(x)=\frac{x^{2}+2}{x^{2}-1}\)

In Exercises 55-62, use a method of your choice to find all horizontal and vertical asymptotes of the function. \(p(x)=\frac{4}{x^{2}+1}\)

In Exercises 55-62, use a method of your choice to find all horizontal and vertical asymptotes of the function. \(g(x)=\frac{4 x-4}{x^{3}-8}\)

In Exercises 55-62, use a method of your choice to find all horizontal and vertical asymptotes of the function. \(h(x)=\frac{2 x-4}{x^{2}-4}\)

In Exercises 55-62, use a method of your choice to find all horizontal and vertical asymptotes of the function. \(y=\frac{x+2}{2 x+1}\)

In Exercises 55-62, use a method of your choice to find all horizontal and vertical asymptotes of the function. \(y=\frac{x^{2}+2}{2 x+1}\)

In Exercises 55-62, use a method of your choice to find all horizontal and vertical asymptotes of the function. \y=\frac{x+2}{2 x^{2}+1}\)

In Exercises 55-62, use a method of your choice to find all horizontal and vertical asymptotes of the function. \(y=\frac{x^{3}+2}{2 x^{2}+1}\)

Can a Graph Cross Its Own Asymptote? The Greek roots of the word asymptote mean not meeting, since graphs tend to approach, but not meet, their asymptotes. Which of the following functions have graphs that do intersect their horizontal asymptotes? (a) \(f(x)=\frac{x}{x^{2}-1}\) (b) \(g(x)=\frac{x}{x^{2}+1}\) (c) \(h(x)=\frac{x^{2}}{x^{3}+1}\)

Can a Graph Have Two Horizontal Asymptotes? Although most graphs have at most one horizontal asymptote, it is possible for a graph to have more than one. Which of the following functions have graphs with more than one horizontal asymptote? (a) \(f(x)=\frac{\left|x^{3}+1\right|}{8-x^{3}}\) (b) \(g(x)=\frac{|x-1|}{x^{2}-4}\) (c) \(h(x)=\frac{x}{\sqrt{x^{2}-4}}\)

Can a Graph Intersect Its Own Vertical Asymptote? Graph the function \(f(x)=\frac{x-|x|}{x^{2}}+1\) (a) The graph of this function does not intersect its vertical asymptote. Explain why it does not. (b) Show how you can add a single point to the graph of and get a graph that does intersect its vertical asymptote. (c) Is the graph in (b) the graph of a function?

Writing to Learn Explain why a graph cannot have more than two horizontal asymptotes.

True or False The graph of function f is defined as the set of all points (x, f(x)), where x is in the domain of f. Justify your answer.

True or False A relation that is symmetric with respect to the x-axis cannot be a function. Justify your answer.

Which function is continuous? (A) Number of children enrolled in a particular school as a function of time (B) Outdoor temperature as a function of time (C) Cost of U.S. postage as a function of the weight of the letter (D) Price of a stock as a function of time (E) Number of soft drinks sold at a ballpark as a function of outdoor temperature

Which function is not continuous? (A) Your altitude as a function of time while flying from Reno to Dallas (B) Time of travel from Miami to Pensacola as a function of driving speed (C) Number of balls that can fit completely inside a particular box as a function of the radius of the balls (D) Area of a circle as a function of radius (E) Weight of a particular baby as a function of time after birth.

Which function is decreasing? (A) Outdoor temperature as a function of time (B) The Dow Jones Industrial Average as a function of time (C) Air pressure in the Earths atmosphere as a function of altitude (D) World population since 1900 as a function of time (E) Water pressure in the ocean as a function of depth

Which function cannot be classified as either increasing or decreasing? (A) Weight of a lead brick as a function of volume (B) Strength of a radio signal as a function of distance from the transmitter (C) Time of travel from Buffalo to Syracuse as a function of driving speed (D) Area of a square as a function of side length (E) Height of a swinging pendulum as a function of time

As promised in Example 7 of this section, we will give you a chance to prove algebraically that \(p(x)=x /\left(1+x^{2}\right)\) is bounded. (a) Graph the function and find the smallest integer k that appears to be an upper bound. (b) Verify that \(x /\left(1+x^{2}\right)<k\) by proving the equivalent inequality \(k x^{2}-x+k>0\). (Use the quadratic formula to show that the quadratic has no real zeros.) (c) From the graph, find the greatest integer k that appears to be a lower bound. (d) Verify that \(x /\left(1+x^{2}\right)>k\) by proving the equivalent inequality \(k x^{2}-x+k<0\).

Baylor School uses a sliding scale to convert the percentage grades on its transcripts to grade point averages (GPAs). Table (1.9 shows the GPA equivalents for selected grades. (a) Considering GPA (y) as a function of percentage grade (x), is it increasing, decreasing, constant, or none of these? (b) Make a table showing the change (\(\Delta\)y) in GPA as you move down the list. (See Exploration (1.) (c) Make a table showing the change in as you move down the list. (This is \(\Delta\)\(\Delta\)y.) Considering the change (\(\Delta\)y) in GPA as a function of percentage grade (x), is it increasing, decreasing, constant, or none of these? (d) In general, what can you say about the shape of the graph if y is an increasing function of x and \(\Delta\)y is a decreasing function of x? (e) Sketch the graph of a function y of x such that y is a decreasing function of x and \(\Delta\)y is an increasing function of x

Sketch (freehand) a graph of a function with domain all real numbers that satisfies all of the following conditions: (a) is continuous for all x; (b) is increasing on \((-\infty, 0]\) and on [3,5] (c) is decreasing on [0,3] and on \([5, \infty)\) (d) ƒ(0) = ƒ(5) = 2; (e) ƒ(3) = 0

Sketch (freehand) a graph of a function with domain all real numbers that satisfies all of the following conditions: (a) f is decreasing on \((-\infty, 0)\) and decreasing on \((0, \infty)\) (b) has a non removable point of discontinuity at x=0 (c) has a horizontal asymptote at y=1 (d) f(0)= 0 (e) f has a vertical asymptote at x= 0

Sketch (freehand) a graph of a function with domain all real numbers that satisfies all of the following conditions: (a)f is continuous for all x; (b) f is an even function; (c)f is increasing on [0,2] and decreasing on [2, \(\infty\) (d) f(2) = 3

Get together with your classmates in groups of two or three. Sketch a graph of a function, but do not show it to the other members of your group. Using the language of functions (as in Exercises 79-81), describe your function as completely as you can. Exchange descriptions with the others in your group and see if you can reproduce each others graphs.

A function that is bounded above has an infinite number of upper bounds, but there is always a least upper bound, i.e., an upper bound that is less than all the others. This least upper bound may or may not be in the range of f. For each of the following functions, find the least upper bound and tell whether or not it is in the range of the function. (a) \(f(x)=2-0.8 x^{2}\) (b) \(g(x)=\frac{3 x^{2}}{3+x^{2}}\) (c) \(h(x)=\frac{1-x}{x^{2}}\) (d) \(p(x)=2 \sin (x)\) (e) \(q(x)=\frac{4 x}{x^{2}+2 x+1}\)

A continuous function f has domain all real numbers. If f(-1)=5 and f(1)=-5, explain why f must have at least one zero in the interval [-1, 1]. (This generalizes to a property of continuous functions known as the Intermediate Value Theorem.)

Proving a Theorem Prove that the graph of every odd function with domain all real numbers must pass through the origin.

Finding the Range Graph the function \(f(x)=\frac{3 x^{2}-1}{2 x^{2}+1}\) in the window [-6,6] by [-2,2]. (a) What is the apparent horizontal asymptote of the graph? (b) Based on your graph, determine the apparent range of f. (c) Show algebraically that \(-1 \leq \frac{3 x^{2}-1}{2 x^{2}+1}<1.5\) for all x, thus confirming your conjecture in part (b).

Looking Ahead to Calculus A key theorem in calculus, the Extreme Value Theorem, states, if a function f is continuous on a closed interval [a, b] then has both a maximum value and a minimum value on the interval. For each of the following functions, verify that the function is continuous on the given interval and find the maximum and minimum values of the function and the x values at which these extrema occur. (a) \(f(x)=x^{2}-3,[-2,4]\) (b) \(f(x)=\frac{1}{x},[1,5]\) (c) \(f(x)=|x+1|+2,[-4,1]\) (d) \(f(x)=\sqrt{x^{2}+9},[-4,4]\)