1. **State the problem:** We have the function $$g(x) = 7 + \frac{6}{5}x - 1$$ with domain restriction $$x \neq 1$$. We need to find: (1a) The images of $$x = -5, -2, 5$$. (1b) Given the image of $$b$$ is $$2b$$, find possible values of $$b$$. --- 2. **Rewrite the function:** Simplify the expression: $$g(x) = 7 + \frac{6}{5}x - 1 = (7 - 1) + \frac{6}{5}x = 6 + \frac{6}{5}x$$ --- 3. **Calculate images for (1a):** - For $$x = -5$$: $$g(-5) = 6 + \frac{6}{5} \times (-5) = 6 + \cancel{\frac{6}{5} \times (-5)} = 6 - 6 = 0$$ - For $$x = -2$$: $$g(-2) = 6 + \frac{6}{5} \times (-2) = 6 - \frac{12}{5} = 6 - 2.4 = 3.6$$ - For $$x = 5$$: $$g(5) = 6 + \frac{6}{5} \times 5 = 6 + 6 = 12$$ --- 4. **Solve for (1b):** Given $$g(b) = 2b$$, find $$b$$. Start with: $$g(b) = 6 + \frac{6}{5}b = 2b$$ Rearranged: $$6 + \frac{6}{5}b = 2b$$ Subtract $$\frac{6}{5}b$$ from both sides: $$6 = 2b - \frac{6}{5}b = \left(2 - \frac{6}{5}\right)b$$ Calculate the coefficient: $$2 - \frac{6}{5} = \frac{10}{5} - \frac{6}{5} = \frac{4}{5}$$ So: $$6 = \frac{4}{5}b$$ Multiply both sides by $$\frac{5}{4}$$: $$b = 6 \times \frac{5}{4} = \frac{30}{4} = 7.5$$ --- **Final answers:** - $$g(-5) = 0$$ - $$g(-2) = 3.6$$ - $$g(5) = 12$$ - Possible value of $$b$$ such that $$g(b) = 2b$$ is $$b = 7.5$$