<< Chapter < Page Chapter >> Page >

The maximum value of a function

In many areas of science, engineering, and mathematics, it is useful to know the maximum value a function can obtain, even if we don’t know its exact value at a given instant. For instance, if we have a function describing the strength of a roof beam, we would want to know the maximum weight the beam can support without breaking. If we have a function that describes the speed of a train, we would want to know its maximum speed before it jumps off the rails. Safe design often depends on knowing maximum values.

This project describes a simple example of a function with a maximum value that depends on two equation coefficients. We will see that maximum values can depend on several factors other than the independent variable x .

  1. Consider the graph in [link] of the function y = sin x + cos x . Describe its overall shape. Is it periodic? How do you know?
    An image of a graph. The x axis runs from -4 to 4 and the y axis runs from -4 to 4. The graph is of the function “y = sin(x) + cos(x)”, a curved wave function. The graph of the function decreases until it hits the approximate point (-(3pi/4), -1.4), where it increases until the approximate point ((pi/4), 1.4), where it begins to decrease again. The x intercepts shown on this graph of the function are at (-(5pi/4), 0), (-(pi/4), 0), and ((3pi/4), 0). The y intercept is at (0, 1).
    The graph of y = sin x + cos x .

    Using a graphing calculator or other graphing device, estimate the x - and y -values of the maximum point for the graph (the first such point where x >0). It may be helpful to express the x -value as a multiple of π.
  2. Now consider other graphs of the form y = A sin x + B cos x for various values of A and B . Sketch the graph when A = 2 and B = 1, and find the x - and y -values for the maximum point. (Remember to express the x -value as a multiple of π, if possible.) Has it moved?
  3. Repeat for A = 1, B = 2. Is there any relationship to what you found in part (2)?
  4. Complete the following table, adding a few choices of your own for A and B :
    A B x y A B x y
    0 1 3 1
    1 0 1 3
    1 1 12 5
    1 2 5 12
    2 1
    2 2
    3 4
    4 3
  5. Try to figure out the formula for the y -values.
  6. The formula for the x -values is a little harder. The most helpful points from the table are ( 1 , 1 ) , ( 1 , 3 ) , ( 3 , 1 ) . ( Hint : Consider inverse trigonometric functions.)
  7. If you found formulas for parts (5) and (6), show that they work together. That is, substitute the x -value formula you found into y = A sin x + B cos x and simplify it to arrive at the y -value formula you found.

Key concepts

  • For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test.
  • If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain.
  • For a function f and its inverse f −1 , f ( f −1 ( x ) ) = x for all x in the domain of f −1 and f −1 ( f ( x ) ) = x for all x in the domain of f .
  • Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions.
  • The graph of a function f and its inverse f −1 are symmetric about the line y = x .

Key equations

  • Inverse functions
    f −1 ( f ( x ) ) = x for all x in D , and f ( f −1 ( y ) ) = y for all y in R .

For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.

For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.

f ( x ) = x 2 4 , x 0

a. f −1 ( x ) = x + 4 b. Domain : x −4 , range : y 0

Got questions? Get instant answers now!

Questions & Answers

questions solve y=sin x
Obi Reply
Solve it for what?
you have to apply the function arcsin in both sides and you get arcsin y = acrsin (sin x) the the function arcsin and function sin cancel each other so the ecuation becomes arcsin y = x you can also write x= arcsin y
what is the question ? what is the answer?
there is an equation that should be solve for x
ok solve it
are you saying y is of sin(x) y=sin(x)/sin of both sides to solve for x... therefore y/sin =x
or solve for sin(x) via the unit circle
what is unit circle
a circle whose radius is 1.
the unit circle is covered in pre cal...and or trigonometry. it is the multipcation table of upper level mathematics.
what is function?
Ryan Reply
A set of points in which every x value (domain) corresponds to exactly one y value (range)
what is lim (x,y)~(0,0) (x/y)
NIKI Reply
limited of x,y at 0,0 is nt defined
But using L'Hopitals rule is x=1 is defined
Could U explain better boss?
value of (x/y) as (x,y) tends to (0,0) also whats the value of (x+y)/(x^2+y^2) as (x,y) tends to (0,0)
can we apply l hospitals rule for function of two variables
why n does not equal -1
K.kupar Reply
ask a complete question if you want a complete answer.
I agree with Andrew
f (x) = a is a function. It's a constant function.
Darnell Reply
proof the formula integration of udv=uv-integration of vdu.?
Bg Reply
Find derivative (2x^3+6xy-4y^2)^2
Rasheed Reply
no x=2 is not a function, as there is nothing that's changing.
Vivek Reply
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
i mean can we replace the roles of x and y and call x=2 as function
if x =y and x = 800 what is y
Joys Reply
how do u factor the numerator?
Drew Reply
Nonsense, you factor numbers
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The problem is the question, is not a problem where it is, but what it is
I think you should first know the basics man: PS
Yes, what factorization is
Antonio bro is x=2 a function?
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
Why y, if domain its usually defined as x, bro, so you creates confusion
Its f(x) =y=2 for every x
Yes but he said could you put x = 2 as a function you put y = 2 as a function
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
The said x=2 and that 2 is y
that 2 is not y, y is a variable 2 is a constant
So 2 is defined as f(x) =2
No y its constant =2
what variable does that function define
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Sorry x=2
And you are right, but os not a function of x, its a function of y
As function of x is meaningless, is not a finction
yeah you mean what I said in my first post, smh
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
OK you can call this "function" on a set {2}, but its a single value function, a constant
well as long as you got there eventually
2x^3+6xy-4y^2)^2 solve this
follow algebraic method. look under factoring numerator from Khan academy
volume between cone z=√(x^2+y^2) and plane z=2
Kranthi Reply
answer please?
It's an integral easy
V=1/3 h π (R^2+r2+ r*R(
How do we find the horizontal asymptote of a function using limits?
Lerato Reply
Easy lim f(x) x-->~ =c
solutions for combining functions
Amna Reply
what is a function? f(x)
Jeremy Reply
one that is one to one, one that passes the vertical line test
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
is x=2 a function?
restate the problem. and I will look. ty
jon Reply
is x=2 a function?
Practice Key Terms 5

Get the best Calculus volume 1 course in your pocket!

Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?