Question on functions and trigonometry.

In summary, the question asks for the inverse function of g(x) when restricted to the domain [-1,1]. The attempt at a solution was to use the quadratic formula to get g^-1(x) = (1 +- root(1-4x^2)) / 2x, but it would be easier to prove that g(x) is always increasing. For the second question, the problem is simplified by rewriting it as sin(2tan^-1(x)) = 2x / (1 + x^2), where x is between -1 and 1.
  • #1
mgnymph
15
0
Sorry, for overloading you guys, but I just can't get my head around these!

Question 1

Homework Statement



Show that, when restricted to the domain [-1,1], g has an inverse function. Use a definition of a theorem, a graph will not be accepted.


Homework Equations



g(x) = x/(x^2 + 1)

The Attempt at a Solution



I tried solving this first by going

y = x/(x^2 + 1)
then making x the subject... so..

yx^2 + y = x
yx^2 - x + y = 0

using quadratic formula, i end up with

g^-1(x) = (1+- root(1 - 4x^2) ) / 2x

But I don't get how to prove its an inverse function when x is between -1 and 1.. :(



Question 2...

Homework Statement



Simplify sin(2tan^-1(x)), tan^-1 is arctan, and state where your simplification is valid.


The Attempt at a Solution



I made a right angled triangle, with adjacent length 1, opposite length x and hypotenuse length root(1 + x^2).

But I don't know how to deal with that 2...



Thanks (in advance) for helping :D
 
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  • #2
Hi mgnymph! :smile:

(have a square-root: √ and a ± and try using the X2 tag just above the Reply box :wink:)
mgnymph said:
Question 1

Homework Statement



Show that, when restricted to the domain [-1,1], g has an inverse function. Use a definition of a theorem, a graph will not be accepted.


Homework Equations



g(x) = x/(x^2 + 1)


yx^2 - x + y = 0

using quadratic formula, i end up with

g^-1(x) = (1+- root(1 - 4x^2) ) / 2x

That's inside-out1 :rolleyes:

You mean x = (1 ± √(1 - 4y2))/2y.

But anyway, wouldn't it be easier just to prove that g(x) is always increasing (by calculus or by a trig substitution)? :wink:
Question 2...

Homework Statement



Simplify sin(2tan^-1(x)), tan^-1 is arctan, and state where your simplification is valid.

Rewrite it …

if y = tanx, then sin2y = … ? :smile:
 
  • #3
tiny-tim said:
Hi mgnymph! :smile:

(have a square-root: √ and a ± and try using the X2 tag just above the Reply box :wink:)


That's inside-out1 :rolleyes:

You mean x = (1 ± √(1 - 4y2))/2y.

But anyway, wouldn't it be easier just to prove that g(x) is always increasing (by calculus or by a trig substitution)? :wink:


Rewrite it …

if y = tanx, then sin2y = … ? :smile:

oh! Thanks for opening my eyes :D
 

What are the basic trigonometric functions?

The basic trigonometric functions are sine, cosine, and tangent. These functions are defined in terms of the sides of a right triangle, where sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.

How are trigonometric functions used in real life?

Trigonometric functions are used in a variety of fields, including engineering, physics, and navigation. They are used to calculate distances, angles, and heights, and are also used in the design of buildings, bridges, and other structures.

What is the unit circle and how is it related to trigonometric functions?

The unit circle is a circle with a radius of 1 unit, centered at the origin on a graph. It is used to define the values of trigonometric functions for any angle. The x-coordinate of a point on the unit circle represents the cosine of the angle, and the y-coordinate represents the sine of the angle.

What is the difference between a function and a relation?

A function is a special type of relation where each input has only one output. In other words, for every x-value, there is only one corresponding y-value. A relation, on the other hand, can have multiple outputs for the same input, making it more general than a function.

What is the domain and range of a function?

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. For trigonometric functions, the domain is typically all real numbers, while the range depends on the specific function and any restrictions on the input values.

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