$\"\"$

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The function is $\"\"$ .

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Determine whether the function $\"\"$ is a one to one or not for all $\"\"$ using horizontal line test.

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Graph the function $\"\"$ .

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$\"\"$

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Observe the graph :

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Draw the horizontal line $\"\"$ , this horizontal line touches the graph twice.

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The function is not an one-to-one, because it violates the horizontal line test.

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$\"\"$

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Consider $\"\"$ .

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Graph the function on the $\"\"$ .

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$\"\"$

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Observe the graph :

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Draw the horizontal line $\"\"$ .

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The horizontal line does not touch the graph more than once.

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The function is a one-to-one, because it passes the horizontal line test.

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$\"\"$

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Assume $\"\"$ .

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Find $\"\"$ .

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If $\"\"$ then $\"\"$ .

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Consider $\"\"$ .

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Differentiate on each side with respect to $\"\"$ .

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$\"\"$

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Trigonometric identity: $\"\"$ .

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$\"\"$

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Therefore, $\"\"$ .

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Substitute $\"\"$ in above expression.

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$\"\"$ .

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$\"\"$

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The function is not an one-to-one for all $\"\"$ .

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The function is a one-to-one on $\"\"$ .

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Derivative of the $\"\"$ is $\"\"$ .