1. **State the problem:** We want to express the function $$R(x) = x - 8$$ as a composition of three functions $$f \circ g \circ h$$, where none of the functions are the identity function. 2. **Recall the composition of functions:** The composition $$f \circ g \circ h$$ means $$f(g(h(x)))$$. 3. **Choose functions:** We want to find functions $$f(x), g(x), h(x)$$ such that $$f(g(h(x))) = x - 8$$. 4. **Stepwise decomposition:** Let's pick $$h(x) = x$$ (not identity since it will be modified later), $$g(x) = x - 8$$, and $$f(x) = x$$. But $$f(x) = x$$ is identity, so we must avoid that. 5. **Alternative approach:** Let $$h(x) = x$$, $$g(x) = x - 8$$, and $$f(x) = x$$ is identity, so no. 6. **Try:** Let $$h(x) = x$$, $$g(x) = x$$, $$f(x) = x - 8$$, but $$g(x) = x$$ is identity. 7. **Try non-identity functions:** - Let $$h(x) = x$$ (identity, no) - Let $$h(x) = x + 1$$ (non-identity) - Then $$g(x) = x - 9$$ (non-identity) - Then $$f(x) = x$$ (identity, no) 8. **Try:** - $$h(x) = x$$ (identity, no) - $$g(x) = x$$ (identity, no) - $$f(x) = x - 8$$ (non-identity) 9. **Try:** - $$h(x) = x$$ (identity, no) - $$g(x) = x - 8$$ (non-identity) - $$f(x) = x$$ (identity, no) 10. **Try:** - $$h(x) = x$$ (identity, no) - $$g(x) = x - 4$$ (non-identity) - $$f(x) = x - 4$$ (non-identity) Then $$f(g(h(x))) = f(g(x)) = f(x - 4) = (x - 4) - 4 = x - 8$$. 11. **Final functions:** - $$h(x) = x$$ (identity, but we must avoid identity) 12. **Adjust h(x):** - Let $$h(x) = x + 1$$ (non-identity) - Then $$g(x) = x - 5$$ (non-identity) - $$f(x) = x - 4$$ (non-identity) Check: $$f(g(h(x))) = f(g(x + 1)) = f((x + 1) - 5) = f(x - 4) = (x - 4) - 4 = x - 8$$. 13. **Summary:** $$h(x) = x + 1$$ $$g(x) = x - 5$$ $$f(x) = x - 4$$ These are all non-identity functions and satisfy $$f(g(h(x))) = x - 8$$. **Answer:** $$\boxed{f(x) = x - 4, \quad g(x) = x - 5, \quad h(x) = x + 1}$$