$\"\"$

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(a)

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The function is $\"\"$ , $\"\"$ and $\"\"$ .

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A function $\"\"$ is said to be one to one if any two elements in the domain are correspond to two different elements in the range.

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If $\"\"$ and $\"\"$ are two different inputs of a function $\"\"$ , then $\"\"$ is said to be one to one provided $\"\"$ .

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If $\"\"$ then

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$\"\"$

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$\"\"$ .

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Therefore, the function $\"\"$ is one-to-one function.

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$\"\"$

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(b)

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The function is $\"\"$ .

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Theorem 7:

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If $\"\"$ is a one to one differentiable function with inverse function $\"\"$ and $\"\"$ then the inverse function is differentiable at $\"\"$ and $\"\"$ .

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Find $\"\"$ .

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Equate the function to $\"\"$ .

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$\"\"$

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$\"\"$ .

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Therefore $\"\"$ then $\"\"$ .

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$\"\"$ .

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Differentiate the function with respect to $\"\"$ .

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Power rule of derivatives : $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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(c)

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The function is $\"\"$ .

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Let $\"\"$ .

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To find the inverse of $\"\"$ , replace $\"\"$ with $\"\"$ and $\"\"$ with $\"\"$ .

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$\"\"$ .

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Solve for $\"\"$ .

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$\"\"$

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$\"\"$ .

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The inverse of the function $\"\"$ is $\"\"$ .

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From the inverse function definition,

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Domain of $\"\"$ is range of $\"\"$ and the range of $\"\"$ is domain of $\"\"$ .

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Domain of $\"\"$ is $\"\"$ .

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For domain $\"\"$ , range of $\"\"$ is $\"\"$ .

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Therefore,

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Domain of $\"\"$ is $\"\"$ .

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Range of $\"\"$ is $\"\"$ .

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$\"\"$

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(d)

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Consider $\"\"$ .

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Differentiate the function with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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Substitute $\"\"$ in above expression.

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$\"\"$ .

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$\"\"$

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(e)

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The graph of $\"\"$ and $\"\"$ is

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$\"\"$

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$\"\"$

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(a) The function $\"\"$ is one-to-one function.

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(b) $\"\"$ .

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(c)

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$\"\"$ .

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Domain of $\"\"$ is $\"\"$ .

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Range of $\"\"$ is $\"\"$ .

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(d) $\"\"$ .

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(e) The graph is

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$\"\"$ .