1. **State the problem:** We are given the function $$f(x) = 3e^{1-x} - 2\sin(\pi x) + x$$ and asked to find the inverse value $$f^{-1}(6)$$, which means we want to find the value of $$x$$ such that $$f(x) = 6$$. 2. **Set up the equation:** To find $$f^{-1}(6)$$, solve $$3e^{1-x} - 2\sin(\pi x) + x = 6$$ 3. **Analyze the equation:** This is a transcendental equation involving exponential, trigonometric, and linear terms. It cannot be solved algebraically in closed form. 4. **Approach:** Use numerical methods or trial and error to approximate $$x$$. 5. **Trial values:** - Try $$x=1$$: $$3e^{1-1} - 2\sin(\pi \cdot 1) + 1 = 3e^0 - 0 + 1 = 3 + 1 = 4$$ (less than 6) - Try $$x=0$$: $$3e^{1-0} - 2\sin(0) + 0 = 3e^{1} + 0 + 0 \approx 3 \times 2.718 = 8.154$$ (greater than 6) 6. **Narrow down:** Since $$f(0) > 6$$ and $$f(1) < 6$$, the solution lies between 0 and 1. 7. **Try $$x=0.5$$:** $$3e^{1-0.5} - 2\sin(\pi \times 0.5) + 0.5 = 3e^{0.5} - 2 \times 1 + 0.5 \approx 3 \times 1.649 - 2 + 0.5 = 4.947 - 2 + 0.5 = 3.447$$ (less than 6) 8. **Try $$x=0.2$$:** $$3e^{1-0.2} - 2\sin(\pi \times 0.2) + 0.2 = 3e^{0.8} - 2 \times \sin(0.2\pi) + 0.2 \approx 3 \times 2.225 - 2 \times 0.588 + 0.2 = 6.675 - 1.176 + 0.2 = 5.699$$ (less than 6) 9. **Try $$x=0.1$$:** $$3e^{1-0.1} - 2\sin(\pi \times 0.1) + 0.1 = 3e^{0.9} - 2 \times \sin(0.1\pi) + 0.1 \approx 3 \times 2.460 - 2 \times 0.309 + 0.1 = 7.38 - 0.618 + 0.1 = 6.862$$ (greater than 6) 10. **Between 0.1 and 0.2:** At $$x=0.1$$, $$f(x) > 6$$; at $$x=0.2$$, $$f(x) < 6$$. 11. **Linear interpolation:** Approximate solution $$x \approx 0.1 + \frac{6 - 6.862}{5.699 - 6.862} \times (0.2 - 0.1) = 0.1 + \frac{-0.862}{-1.163} \times 0.1 \approx 0.1 + 0.074 = 0.174$$ 12. **Final answer:** $$f^{-1}(6) \approx 0.174$$ This means when $$x \approx 0.174$$, $$f(x) = 6$$.