Solve Inverse Trig Function with Domain [-π/2, π/2] | Step-by-Step Guide

In summary, the problem is asking for the inverse of an equation which has sin(x) in it. For this, you would use the fact that \arcsin(x) = y to solve for x.
  • #1
togame
18
0

Homework Statement


My problem is as follows:
find the inverse of
[tex] 3x+1+\sin(x)[/tex] with the domain [itex][-\frac{\pi}{2},\frac{\pi}{2}][/itex]


Homework Equations





The Attempt at a Solution


for this would I just try to solve as normal by setting y=f(x) then using the fact that [itex]\arcsin(x) = y[/itex] or is this the wrong way of solving this?
 
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  • #2
togame said:

Homework Statement


My problem is as follows:
find the inverse of
[tex] 3x+1+\sin(x)[/tex] with the domain [itex][-\frac{\pi}{2},\frac{\pi}{2}][/itex]


Homework Equations





The Attempt at a Solution


for this would I just try to solve as normal by setting y=f(x) then using the fact that [itex]\arcsin(x) = y[/itex] or is this the wrong way of solving this?

You're not going to be able to solve the equation y = f(x) = 3x + 1 + sin(x) for x (to get the inverse x = f-1(y).
What is the exact problem statement? It might be that you are misreading what is being asked for in this problem.
 
  • #3
The exact wording for this problem:
[itex]f(x) = 3x + 1 + \sin(x)[/itex] with domain [itex][-\pi/2, \pi/2][/itex]. Without your calculator, determine the value of [itex]f^{-1}(1)[/itex].

Since this is an inverse function, I was going to try to solve for the inverse function, then solve for [itex]f^{-1}(1)[/itex]
 
  • #4
How is this question related to a problem which requires you to:
Solve [itex]f(x)=1\text{ for }x\,.[/itex]​
?
 
  • #5
SammyS said:
How is this question related to a problem which requires you to:
Solve [itex]f(x)=1\text{ for }x\,.[/itex]​
?

To which question are you referring? I never mentioned having to solve [itex]f(x)=1[/itex]
 
  • #6
The problem asks to determine f-1(1)=x. The equation is equivalent to 1 =f(x), that is, 3x+1+sin(x)=1. Solve for x.

ehild
 
  • #7
SammyS said:
How is this question related to a problem which requires you to:
Solve [itex]f(x)=1\text{ for }x\,.[/itex]​
?

togame said:
To which question are you referring? I never mentioned having to solve [itex]f(x)=1[/itex]
I'll state it more clearly.

How are the following two problems related?
Determine the value of [itex]f^{-1}(1)\,.[/itex]

Solve [itex]f(x)=1\text{ for }x\,.[/itex]​
 
  • #8
togame said:
The exact wording for this problem:
[itex]f(x) = 3x + 1 + \sin(x)[/itex] with domain [itex][-\pi/2, \pi/2][/itex]. Without your calculator, determine the value of [itex]f^{-1}(1)[/itex].

Since this is an inverse function, I was going to try to solve for the inverse function, then solve for [itex]f^{-1}(1)[/itex]
While that is the general method, if you are lucky the problem is one that can be solved by recognizing it to be a special case that is easy to solve without ploughing all the way through the full general method (even if the general method were possible).

So you are being asked to find the x value (or values) that makes

[itex]3x + 1 + \sin(x) = 1[/itex]


Play around with that equation to see whether you can knock it into something that speaks meaningfully to you. :smile:
 
  • #9
SammyS said:
I'll state it more clearly.

How are the following two problems related?
Determine the value of [itex]f^{-1}(1)\,.[/itex]

Solve [itex]f(x)=1\text{ for }x\,.[/itex]​

NascentOxygen said:
While that is the general method, if you are lucky the problem is one that can be solved by recognizing it to be a special case that is easy to solve without ploughing all the way through the full general method (even if the general method were possible).

So you are being asked to find the x value (or values) that makes

[itex]3x + 1 + \sin(x) = 1[/itex]


Play around with that equation to see whether you can knock it into something that speaks meaningfully to you. :smile:

Ah, I see what you guys are talking about now. Thanks for the help. Much appreciated.
 

1. What is an inverse trig function?

An inverse trig function is the inverse of a trigonometric function. It helps to find the angle or angles that satisfy a given trigonometric equation.

2. What does the given domain mean?

The given domain, [-π/2, π/2], means that the angle or angles we are looking for must be between -π/2 and π/2. This is because the inverse trig functions have a restricted range of values that they can output.

3. How do I solve an inverse trig function with the given domain?

To solve an inverse trig function with the given domain, you can follow these steps:

  1. Write the inverse trig function in the form of y = arctan(x), arcsin(x), or arccos(x).
  2. Set y equal to the given value.
  3. Take the inverse of the trig function to isolate x.
  4. Check if the angle or angles found are within the given domain.

4. Can I use a calculator to solve an inverse trig function?

Yes, most scientific calculators have inverse trig functions and can be used to solve these types of equations. However, it is important to make sure the calculator is set to the correct mode (degrees or radians) and that the values are within the given domain.

5. What should I do if the angle or angles I find are not within the given domain?

If the angle or angles you find are not within the given domain, it means that there is no solution to the inverse trig function with the given domain. You may need to use a different method or approach to solve the problem.

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