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Rule: horizontal line test

A function f is one-to-one if and only if every horizontal line intersects the graph of f no more than once.

An image of two graphs. Both graphs have an x axis that runs from -3 to 3 and a y axis that runs from -3 to 4. The first graph is of the function “f(x) = x squared”, which is a parabola. The function decreases until it hits the origin, where it begins to increase. The x intercept and y intercept are both at the origin. There are two orange horizontal lines also plotted on the graph, both of which run through the function at two points each. The second graph is of the function “f(x) = x cubed”, which is an increasing curved function. The x intercept and y intercept are both at the origin. There are three orange lines also plotted on the graph, each of which only intersects the function at one point.
(a) The function f ( x ) = x 2 is not one-to-one because it fails the horizontal line test. (b) The function f ( x ) = x 3 is one-to-one because it passes the horizontal line test.

Determining whether a function is one-to-one

For each of the following functions, use the horizontal line test to determine whether it is one-to-one.

  1. An image of a graph. The x axis runs from -3 to 11 and the y axis runs from -3 to 11. The graph is of a step function which contains 10 horizontal steps. Each steps starts with a closed circle and ends with an open circle. The first step starts at the origin and ends at the point (1, 0). The second step starts at the point (1, 1) and ends at the point (1, 2). Each of the following 8 steps starts 1 unit higher in the y direction than where the previous step ended. The tenth and final step starts at the point (9, 9) and ends at the point (10, 9)
  2. An image of a graph. The x axis runs from -3 to 6 and the y axis runs from -3 to 6. The graph is of the function “f(x) = (1/x)”, a curved decreasing function. The graph of the function starts right below the x axis in the 4th quadrant and begins to decreases until it comes close to the y axis. The graph keeps decreasing as it gets closer and closer to the y axis, but never touches it due to the vertical asymptote. In the first quadrant, the graph of the function starts close to the y axis and keeps decreasing until it gets close to the x axis. As the function continues to decreases it gets closer and closer to the x axis without touching it, where there is a horizontal asymptote.
  1. Since the horizontal line y = n for any integer n 0 intersects the graph more than once, this function is not one-to-one.
    An image of a graph. The x axis runs from -3 to 11 and the y axis runs from -3 to 11. The graph is of a step function which contains 10 horizontal steps. Each steps starts with a closed circle and ends with an open circle. The first step starts at the origin and ends at the point (1, 0). The second step starts at the point (1, 1) and ends at the point (1, 2). Each of the following 8 steps starts 1 unit higher in the y direction than where the previous step ended. The tenth and final step starts at the point (9, 9) and ends at the point (10, 9). There are also two horizontal orange lines plotted on the graph, each of which run through an entire step of the function.
  2. Since every horizontal line intersects the graph once (at most), this function is one-to-one.
    An image of a graph. The x axis runs from -3 to 6 and the y axis runs from -3 to 6. The graph is of the function “f(x) = (1/x)”, a curved decreasing function. The graph of the function starts right below the x axis in the 4th quadrant and begins to decreases until it comes close to the y axis. The graph keeps decreasing as it gets closer and closer to the y axis, but never touches it due to the vertical asymptote. In the first quadrant, the graph of the function starts close to the y axis and keeps decreasing until it gets close to the x axis. As the function continues to decreases it gets closer and closer to the x axis without touching it, where there is a horizontal asymptote. There are also three horizontal orange lines plotted on the graph, each of which only runs through the function at one point.
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Is the function f graphed in the following image one-to-one?

An image of a graph. The x axis runs from -3 to 4 and the y axis runs from -3 to 5. The graph is of the function “f(x) = (x cubed) - x” which is a curved function. The function increases, decreases, then increases again. The x intercepts are at the points (-1, 0), (0,0), and (1, 0). The y intercept is at the origin.

No.

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Finding a function’s inverse

We can now consider one-to-one functions and show how to find their inverses. Recall that a function maps elements in the domain of f to elements in the range of f . The inverse function maps each element from the range of f back to its corresponding element from the domain of f . Therefore, to find the inverse function of a one-to-one function f , given any y in the range of f , we need to determine which x in the domain of f satisfies f ( x ) = y . Since f is one-to-one, there is exactly one such value x . We can find that value x by solving the equation f ( x ) = y for x . Doing so, we are able to write x as a function of y where the domain of this function is the range of f and the range of this new function is the domain of f . Consequently, this function is the inverse of f , and we write x = f −1 ( y ) . Since we typically use the variable x to denote the independent variable and y to denote the dependent variable, we often interchange the roles of x and y , and write y = f −1 ( x ) . Representing the inverse function in this way is also helpful later when we graph a function f and its inverse f −1 on the same axes.

Problem-solving strategy: finding an inverse function

  1. Solve the equation y = f ( x ) for x .
  2. Interchange the variables x and y and write y = f −1 ( x ) .

Finding an inverse function

Find the inverse for the function f ( x ) = 3 x 4 . State the domain and range of the inverse function. Verify that f −1 ( f ( x ) ) = x .

Follow the steps outlined in the strategy.

Step 1. If y = 3 x 4 , then 3 x = y + 4 and x = 1 3 y + 4 3 .

Step 2. Rewrite as y = 1 3 x + 4 3 and let y = f −1 ( x ) .

Therefore, f −1 ( x ) = 1 3 x + 4 3 .

Since the domain of f is ( , ) , the range of f −1 is ( , ) . Since the range of f is ( , ) , the domain of f −1 is ( , ) .

You can verify that f −1 ( f ( x ) ) = x by writing

f −1 ( f ( x ) ) = f −1 ( 3 x 4 ) = 1 3 ( 3 x 4 ) + 4 3 = x 4 3 + 4 3 = x .

Note that for f −1 ( x ) to be the inverse of f ( x ) , both f −1 ( f ( x ) ) = x and f ( f −1 ( x ) ) = x for all x in the domain of the inside function.

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Find the inverse of the function f ( x ) = 3 x / ( x 2 ) . State the domain and range of the inverse function.

f −1 ( x ) = 2 x x 3 . The domain of f −1 is { x | x 3 } . The range of f −1 is { y | y 2 } .

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Graphing inverse functions

Let’s consider the relationship between the graph of a function f and the graph of its inverse. Consider the graph of f shown in [link] and a point ( a , b ) on the graph. Since b = f ( a ) , then f −1 ( b ) = a . Therefore, when we graph f −1 , the point ( b , a ) is on the graph. As a result, the graph of f −1 is a reflection of the graph of f about the line y = x .

Questions & Answers

The f'(4)for f(x) =4^x
Alice Reply
I need help under implicit differentiation
Uchenna Reply
how to understand this
nhie
understand what?
Boniface
how dont you understand it is obvious
Se
if it's there, it must be differentiable
Giorgio
marginal rate of substitution in economics for an example is an implicit differentiation function. It is a proportion of comparison. It could be expressed as the area of a triangle of the old value proportional to the new, and then the next value and so on.
James
The new shapes form a line with a derivative curve
James
That curve could be expressed mathematically. A good real life example is the proportion at which people barter in pawn shops. "How about 100?" "What are you trying to do? rob me? 50!" "No way chap. 75." "Ill give you 62. deal?" "68. Nothing more nothing less." "deal"
James
the proportion of what someone was trying to get for their product, versus what they were offered to the new price they wanted for their product and what they were offered
James
The proportion of the differentiating triangles would be somewhat 1:2. and since there is little variation to the curve then it looks more like a straight line.
James
By the way This is one of the hardest subjects for me. I have a really hard time expressing things in such a way. I'm trying to have more exact calculations which is why I still study the subject.
James
What is derivative of antilog x dx ?
Tanmay Reply
what's the meaning of removable discontinuity
Brian Reply
what's continuous
Brian
an area under a curve is continuous because you are looking at an area that covers a range of numbers, it is over an interval, such as 0 to 4
Lauren
using product rule x^3,x^5
Kabiru
please help me to calculus
World Reply
may god be with you
Sunny
Luke 17:21 nor will they say, See here or See there For indeed, the kingdom of God is within you. You've never 'touched' anything. The e-energy field created by your body has pushed other electricfields. even our religions tell us we're the gods. We live in energies connecting us all. Doa/higgsfield
Scott
if you have any calculus questions many of us would be happy to help and you can always learn or even invent your own theories and proofs. math is the laws of logic and reality. its rules are permanent and absolute. you can absolutely learn calculus and through it better understand our existence.
Scott
ya doubtless
Bilal
help the integral of x^2/lnxdx
Levis
also find the value of "X" from the equation that follow (x-1/x)^4 +4(x^2-1/x^2) -6=0 please guy help
Levis
Use integration by parts. Let u=lnx and dv=x2dx Then du=1xdx and v=13x3. ∫x2lnxdx=13x3lnx−∫(13x3⋅1x)dx ∫x2lnxdx=13x3lnx−∫13x2dx ∫x2lnxdx=13x3lnx−19x3+C
Bilal
itz 1/3 and 1/9
Bilal
now you can find the value of X from the above equation easily
Bilal
Pls i need more explanation on this calculus
usman
usman from where do you need help?
Levis
thanks Bilal
Levis
integrate e^cosx
Uchenna
-sinx e^x
Leo
Do we ask only math question? or ANY of the question?
Levis Reply
yh
Gbesemete
How do i differentiate between substitution method, partial fraction and algebraic function in integration?
usman
you just have to recognize the problem. there can be multiple ways to solve 1 problem. that's the hardest part about integration
Lauren
test
MOHAMMAD
we asking the question cause only the question will tell us the right answer
Sunny
find integral of sin8xcos12xdx
Levis Reply
don't share these childish questions
Bilal
well find the integral of x^x
Levis
bilal kumhar you are so biased if you are an expert what are you doing here lol😎😎😂😂 we are here to learn and beside there are many questions on this chat which you didn't attempt we are helping each other stop being naive and arrogance so give me the integral of x^x
Levis
Levis I am sorry
Bilal
Bilal it okay buddy honestly i am pleasured to meet you
Levis
x^x ... no anti derivative for this function... but we can find definte integral numerically.
Bilal
thank you Bilal Kumhar then how we may find definite integral let say x^x,3,5?
Levis
evaluate 5-×square divided by x+2 find x as limit approaches infinity
Michagaye Reply
i have not understood
Leo
The answer is 0
Michael
welcome
Sunny
I just dont get it at all...not understanding
Michagaye
0 baby
Sunny
The denominator is the aggressive one
Sunny
wouldn't be any prime number for x instead ?
Harold
or should I say any prime number greater then 11 ?
Harold
just wondering
Harold
I think as limit Approach infinity then X=0
Levis
ha hakdog hahhahahaha
No Reply
ha hamburger
Leonito
Fond the value of the six trigonometric function of an angle theta, which terminal side passes through the points(2x½-y)²,4
albert Reply
What's f(x) ^x^x
Emeka Reply
What's F(x) =x^x^x
Emeka
are you asking for the derivative
Leo
that's means more power for all points
rd
if your asking for derivative dy/dz=x^2/2(lnx-1/2)
Levis
iam sorry f(x)=x^x it means the output(range ) depends to input(domain) value of x by the power of x that is to say if x=2 then x^x would be 2^2=4 f(x) is the product of X to the power of X its derivatives is found by using product rule y=x^x introduce ln each side we have lny=lnx^x =lny=xlnx
Levis
the derivatives of f(x)=x^x IS (1+lnx)*x^x
Levis
So in that case what will be the answer?
Alice
nice explanation Levis, appreciated..
Thato
what is a maximax
Chinye Reply
A maxima in a curve refers to the maximum point said curve. The maxima is a point where the gradient of the curve is equal to 0 (dy/dx = 0) and its second derivative value is a negative (d²y/dx² = -ve).
Viewer
what is the limit of x^2+x when x approaches 0
Dike Reply
it is 0 because 0 squared Is 0
Leo
0+0=0
Leo
simply put the value of 0 in places of x.....
Tonu
the limit is 2x + 1
Nicholas
the limit is 0
Muzamil
limit s x
Bilal
The limit is 3
Levis
Leo we don't just do like that buddy!!! use first principle y+∆y=x+∆x ∆y=x+∆x-y ∆y=(x+∆x)^2+(x+∆x)-x^2+x on solving it become ∆y=3∆x+∆x^2 as ∆x_>0 limit=3 if you do by calculator say plugging any value of x=0.000005 which approach 0 you get 3
Levis
find derivatives 3√x²+√3x²
Care Reply
3 + 3=6
mujahid
How to do basic integrals
dondi Reply
the formula is simple x^n+1/n+1 where n IS NOT EQUAL TO 1 And n stands for power eg integral of x^2 x^2+1/2+1 =X^3/3
Levis
Practice Key Terms 5

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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