Chapter 7: Q. 12 (page 450)

True or False $y = \tan^{- 1} x m e a n s x = \tan y, w h e r e - \infty < x < \infty a n d - \frac{π}{2} < y < \frac{π}{2}$

Short Answer

Expert verified

True statement : $y = \tan^{- 1} x m e a n s x = \tan y, w h e r e - \infty < x < \infty a n d - \frac{π}{2} < y < \frac{π}{2}$

Step by step solution

Step 1. Given Information:

Given Statement:

$y = \tan^{- 1} x m e a n s x = \tan y, w h e r e - \infty < x < \infty a n d - \frac{π}{2} < y < \frac{π}{2}$

We want to know is the given statement is True or False.

Step 2. Solution:

The tan^-1 function is the inverse of the tan function so they cancel each other under predefined conditions.

We know that

The range of the function = Domain of the original function.
The domain of the inverse = range of the original function.

Applying the tan function both sides of the given expression we get:

$\tan (y) = \tan (\tan^{- 1} x)$

Therefore, we get:

$x = \tan y, w h e r e - \infty < x < \infty a n d - \frac{π}{2} < y < \frac{π}{2}$

Hence, the statements is True.

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True or False $y = \tan^{- 1} x m e a n s x = \tan y, w h e r e - \infty < x < \infty a n d - \frac{π}{2} < y < \frac{π}{2}$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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True or Falsey=tan-1xmeansx=tany,where-∞<x<∞and-π2<y<π2

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Step by step solution

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True or False $y = \tan^{- 1} x m e a n s x = \tan y, w h e r e - \infty < x < \infty a n d - \frac{π}{2} < y < \frac{π}{2}$