# Write a range of $f\left( x \right)={{\sin }^{-1}}x$ other than $\left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]$

Answer

Verified

280.8k+ views

**Hint:**To find the range of given function other than the given range we will plot its graph and check its value in other points. Firstly we will draw a graph of a given function then we will see under what range its value falls. Then we will take a range accordingly and get our desired answer.

**Complete step by step answer:**

The function given to us is as follows:

$f\left( x \right)={{\sin }^{-1}}x$

The range is provided to us as follows:

$\left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]$

We will find value of the function in this range as follows:

So at $x=-\dfrac{\pi }{2}$ we get the value as:

$\begin{align}

& f\left( \dfrac{\pi }{2} \right)={{\sin }^{-1}}\left( -\dfrac{\pi }{2} \right) \\

& \Rightarrow f\left( \dfrac{\pi }{2} \right)=-{{\sin }^{-1}}\left( \dfrac{\pi }{2} \right) \\

& \Rightarrow f\left( \dfrac{\pi }{2} \right)=-1 \\

\end{align}$

So at $x=\dfrac{\pi }{2}$ we get the value as:

$\begin{align}

& f\left( \dfrac{\pi }{2} \right)={{\sin }^{-1}}\left( \dfrac{\pi }{2} \right) \\

& \Rightarrow f\left( \dfrac{\pi }{2} \right)=1 \\

\end{align}$

So the value of function $f\left( x \right)={{\sin }^{-1}}x$ lies in $\left[ -1,1 \right]$

We get the graph of the function as below:

So we have to take a range that gives the value under the above graph.

So we can take the range as $\left[ \dfrac{\pi }{2},\dfrac{3\pi }{2} \right]$

Hence range of $f\left( x \right)={{\sin }^{-1}}x$ other than $\left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]$ is $\left[ \dfrac{\pi }{2},\dfrac{3\pi }{2} \right]$

**Note:**Trigonometric is a branch of mathematics that studies the relation between the side lengths and the angles of a triangle. There are six types of trigonometric functions which are sine, cosine, tangent, secant, cosecant and cotangent. As the six trigonometric functions are periodic in nature they are not injective and hence they are invertible by restricting the domain of the function. The graph of the inverse of the sine function is like a reflection over the line $y=x$ of the sine function. Sometimes we write the inverse function as $\arcsin \left( x \right)$ because the superscript $-1$ is not an exponent so to avoid any confusion a different notation can be used.

Recently Updated Pages

Basicity of sulphurous acid and sulphuric acid are

Why should electric field lines never cross each other class 12 physics CBSE

An electrostatic field line is a continuous curve That class 12 physics CBSE

What are the measures one has to take to prevent contracting class 12 biology CBSE

Suggest some methods to assist infertile couples to class 12 biology CBSE

Amniocentesis for sex determination is banned in our class 12 biology CBSE

Trending doubts

What is 1 divided by 0 class 8 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is the past tense of read class 10 english CBSE

What is pollution? How many types of pollution? Define it

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

How many crores make 10 million class 7 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE