XERCISE 2 Deternine the equation of the inverse of each of the following functions and then sketch the function and its inverse on the same set of axes, showing the line of symmetry: (1) \( f(x)=3 x-6 \) (2) \( y=2 x^{2} \) (3) \( g(x)=3^{1} \) (d) \( y=-x^{2} \) (5) \( \quad h(x)=\left(\frac{1}{3}\right)^{1} \) (6) \( \quad g(x)=\log _{2} x \) (7) \( y=-\frac{1}{2} x^{2} \) (8) \( f(x)=4=x \) (9) \( \quad h(x)=\log _{1} x \) \( (10)^{*} \quad f(x)=-3^{1} \) \( (11)^{*} \quad h(x)=-\log _{y} x \) \( (12)^{*} \quad g(x)=\log _{2}(-x) \)

To find the inverse of each function, we usually switch the roles of \(x\) and \(y\) and then solve for \(y\): 1. For \( f(x) = 3x - 6 \), switch to get \( x = 3y - 6 \). Solving yields the inverse: \( f^{-1}(x) = \frac{x + 6}{3} \). 2. For \( y = 2x^{2} \), since it's not one-to-one, we need to restrict the domain. Assuming \( x \geq 0 \), the inverse is \( y = \sqrt{\frac{x}{2}} \). 3. The function \( g(x) = 3^1 \) gives a constant output of 3, which doesn't have an inverse as it fails the horizontal line test. 4. For \( y = -x^{2} \), again a not one-to-one function. If we restrict \( x \leq 0 \), we find the inverse would be \( y = -\sqrt{-x} \). 5. For \( h(x) = \left( \frac{1}{3} \right)^1 \), like \( g(x) \), it is a constant value and thus has no inverse. 6. For \( g(x) = \log_{2} x \), the inverse would be \( g^{-1}(x) = 2^x \). 7. The function \( y = -\frac{1}{2} x^{2} \) also needs a restricted domain. For \( x \leq 0 \), its inverse would be \( y = -\sqrt{-2x} \). 8. The function \( f(x) = 4 = x \) indicates \( x \) is always 4, leading to no inverse as it's a constant. 9. The function \( h(x) = \log_{1} x \) is incorrect because the logarithm with base 1 is undefined. 10. For \( f(x) = -3^1 \), similarly, this results in an output of -3 with no inverse. 11. The function \( h(x) = -\log_{y} x \) requires a specified base \(y\); without one, we can't determine the inverse. 12. Lastly, \( g(x) = \log_{2}(-x) \) is valid for \( x < 0 \), yielding \( g^{-1}(x) = -2^x \). For sketching, draw each function with its respective inverse on a coordinate plane, highlighting the line \(y = x\) as the line of symmetry!