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) $$ \quad h(x)=\log _{1} x $$ $$ (10)^{*} \quad f(x)=-3^{1} $$ $$ (11)^{*} \quad h(x)=-\log _{y} x $$ $$ (12)^{*} \quad g(x)=\log _{2}(-x) $$

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To determine the equations of the inverses for the given functions, we will follow these steps for each function:

1. Replace $$ f(x) $$ with $$ y $$.
2. Swap $$ x $$ and $$ y $$.
3. Solve for $$ y $$ to find the inverse function.
4. Sketch the original function and its inverse on the same axes, along with the line of symmetry $$ y = x $$.

Let's go through each function step by step.

### (1) $$ f(x) = 3x - 6 $$

1. Replace $$ f(x) $$ with $$ y $$:
   $$
   y = 3x - 6
   $$
2. Swap $$ x $$ and $$ y $$:
   $$
   x = 3y - 6
   $$
3. Solve for $$ y $$:
   $$
   3y = x + 6 \implies y = \frac{x + 6}{3}
   $$
   Thus, the inverse is:
   $$
   f^{-1}(x) = \frac{x + 6}{3}
   $$

### (2) $$ y = 2x^2 $$

1. Replace $$ y $$:
   $$
   y = 2x^2
   $$
2. Swap $$ x $$ and $$ y $$:
   $$
   x = 2y^2
   $$
3. Solve for $$ y $$:
   $$
   y^2 = \frac{x}{2} \implies y = \sqrt{\frac{x}{2}} \quad (\text{considering } y \geq 0)
   $$
   Thus, the inverse is:
   $$
   f^{-1}(x) = \sqrt{\frac{x}{2}}
   $$

### (3) $$ g(x) = 3^x $$

1. Replace $$ g(x) $$:
   $$
   y = 3^x
   $$
2. Swap $$ x $$ and $$ y $$:
   $$
   x = 3^y
   $$
3. Solve for $$ y $$:
   $$
   y = \log_3 x
   $$
   Thus, the inverse is:
   $$
   g^{-1}(x) = \log_3 x
   $$

### (4) $$ y = -x^2 $$

1. Replace $$ y $$:
   $$
   y = -x^2
   $$
2. Swap $$ x $$ and $$ y $$:
   $$
   x = -y^2
   $$
3. Solve for $$ y $$:
   $$
   -y^2 = x \implies y^2 = -x \implies y = \sqrt{-x} \quad (\text{considering } y \leq 0)
   $$
   Thus, the inverse is:
   $$
   f^{-1}(x) = -\sqrt{-x}
   $$

### (5) $$ h(x) = \left(\frac{1}{3}\right)^x $$

1. Replace $$ h(x) $$:
   $$
   y = \left(\frac{1}{3}\right)^x
   $$
2. Swap $$ x $$ and $$ y $$:
   $$
   x = \left(\frac{1}{3}\right)^y
   $$
3. Solve for $$ y $$:
   $$
   y = -\log_{\frac{1}{3}} x
   $$
   Thus, the inverse is:
   $$
   h^{-1}(x) = -\log_{\frac{1}{3}} x
   $$

### (6) $$ g(x) = \log_2 x $$

1. Replace $$ g(x) $$:
   $$
   y = \log_2 x
   $$
2. Swap $$ x $$ and $$ y $$:
   $$
   x = \log_2 y
   $$
3. Solve for $$ y $$:
   $$
   y = 2^x
   $$
   Thus, the inverse is:
   $$
   g^{-1}(x) = 2^x
   $$

### (7) $$ y = -\frac{1}{2} x^2 $$

1. Replace $$ y $$:
   $$
   y = -\frac{1}{2} x^2
   $$
2. Swap $$ x $$ and $$ y $$:
   $$
   x = -\frac{1}{2} y^2
   $$
3. Solve for $$ y $$:
   $$
   -\frac{1}{2} y^2 = x \implies y^2 = -2x \implies y = -\sqrt{-2x} \quad (\text{considering } y \leq 0)
   $$
   Thus, the inverse is:
   $$
   f^{-1}(x) = -\sqrt{-2x}
   $$

### (8) $$ f(x) = 4 = x $$

This function is not a function of $$ x $$ but rather a constant. It does not have an inverse.

### (9) $$ h(x) = \log_1 x $$

This function is not defined since the base of a logarithm cannot be 1. It does not have an inverse.

### (10) $$ f(x) = -3^x $$

1. Replace $$ f(x) $$:
   $$
   y = -3^x
   $$
2. Swap $$ x $$ and $$ y $$:
   $$
   x = -3^y
   $$
3. Solve for $$ y $$:
   $$
   -3^y = x \implies 3^y = -x \quad (\text{not valid for real } x)
   $$
   Thus, this function does not have a real inverse.

### (11) $$ h(x) = -\log_y x $$

This function is not defined since the base of a logarithm must be positive and not equal to 1. It does not have an inverse.

### (12) $$ g(x) = \log_2(-x) $$

1. Replace $$ g(x) $$:
   $$
   y = \log_2(-x)
   $$
2. Swap $$ x $$ and $$ y $$:
   $$
   x = \log_2(-y)
   $$
3. Solve for $$ y $$:
   $$
   -y = 2^x \implies y = -2^x
   $$
   Thus, the inverse is:
   $$
   g^{-1}(x) = -2^x
   $$

### Summary of Inverses

1. $$ f^{-1}(x) = \frac{x + 6}{3} $$
2. $$ f^{-1}(x) = \sqrt{\frac{x}{2}} $$
3
Here are the inverse functions for the given functions: 1. $$ f^{-1}(x) = \frac{x + 6}{3} $$ 2. $$ f^{-1}(x) = \sqrt{\frac{x}{2}} $$ 3. $$ g^{-1}(x) = \log_3 x $$ 4. $$ f^{-1}(x) = -\sqrt{-x} $$ 5. $$ h^{-1}(x) = -\log_{\frac{1}{3}} x $$ 6. $$ g^{-1}(x) = 2^x $$ 7. $$ f^{-1}(x) = -\sqrt{-2x} $$ 8. No inverse exists. 9. No inverse exists. 10. No real inverse exists. 11. No inverse exists. 12. $$ g^{-1}(x) = -2^x $$

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