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The function $g$ is defined as $g(x) = \ln(x^2), x \in \mathbb{R}, x < 0$ . (i) State the domain and range of $g$ . (ii) Give a reason why $g^{-1}$ exists. (iii) Find the rule, domain and range of $g^{-1}$ .

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Find derivatives of other trigonometric functions

Full solution

Q. The function

g

is defined as

g(x) = \ln(x^2), x \in \mathbb{R}, x < 0

. (i) State the domain and range of

g

. (ii) Give a reason why

g^{-1}

exists. (iii) Find the rule, domain and range of

g^{-1}

.

Define function $g$ : The function $g$ is defined as $g(x) = \ln(x^2)$ for $x < 0$ . To find the domain of $g$ , we need to consider the values of $x$ for which $\ln(x^2)$ is defined. $\newline$ Since the natural logarithm function, $\ln(x)$ , is defined for $x > 0$ , and $x^2$ is always positive for $g$ $0$ , the domain of $g$ is all negative real numbers.
Find domain of $\newline$ $g$ : The range of $\newline$ $g$ is the set of all possible output values. Since the natural logarithm function can take any real number as its output, and $\newline$ $x^2$ maps any negative $\newline$ $x$ to a positive number, the range of $\newline$ $g$ is all real numbers.
Find range of $g$ : To give a reason why the inverse of $g$ exists, we need to check if $g$ is a one-to-one function. A function is one-to-one if each output is the result of exactly one input. Since the derivative of $g(x) = \ln(x^2)$ is $g'(x) = \frac{2}{x}$ , which is always negative for $x < 0$ , $g$ is strictly decreasing on its domain. Therefore, $g$ is one-to-one and an inverse function $g^{-1}$ exists.
Check one-to-one function: To find the rule of the inverse function $g^{-1}$ , we set $y = \ln(x^2)$ and solve for $x$ in terms of $y$ . We exponentiate both sides to remove the natural logarithm, getting $e^y = x^2$ . Since we know $x$ is negative (from the domain of $g$ ), we take the negative square root to find $x$ . Thus, $x = -\sqrt{e^y}$ , and the rule for $g^{-1}$ is $y = \ln(x^2)$ $0$ .
Find rule of inverse function: The domain of the inverse function $g^{-1}$ is the range of the original function $g$ , which is all real numbers, as we have established earlier.
Domain of inverse function: The range of the inverse function $g^{-1}$ is the domain of the original function $g$ , which is all negative real numbers.

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