1. **Problem statement:** We are asked to explain what a function is and why the composition of two functions is still a function. Then, we will solve the first example: given $f_{10}(v) = \frac{1}{v}$ and $g_{10}(w) = w^2 + 2w + 4$, find the composition $f_{10}(g_{10}(x))$. 2. **What is a function?** A function is a rule that assigns exactly one output to each input from its domain. This means for every input value, there is one and only one output value. 3. **Composition of functions:** When we compose two functions, say $f$ and $g$, the output of $g$ becomes the input of $f$. The composition $f(g(x))$ is still a function because for each input $x$, $g(x)$ produces a unique output, and then $f$ takes that unique output and produces another unique output. So the overall rule still assigns exactly one output to each input. 4. **Example:** Given $$f_{10}(v) = \frac{1}{v}$$ $$g_{10}(w) = w^2 + 2w + 4$$ We want to find $$f_{10}(g_{10}(x)) = f_{10}\bigl(g_{10}(x)\bigr) = f_{10}\bigl(x^2 + 2x + 4\bigr)$$ 5. **Substitute $g_{10}(x)$ into $f_{10}$:** $$f_{10}(g_{10}(x)) = \frac{1}{x^2 + 2x + 4}$$ 6. **Final answer:** $$\boxed{f_{10}(g_{10}(x)) = \frac{1}{x^2 + 2x + 4}}$$ This is the composed function. It is still a function because the denominator $x^2 + 2x + 4$ is defined for all real $x$ (it never equals zero since its discriminant $\Delta = 2^2 - 4 \cdot 1 \cdot 4 = 4 - 16 = -12 < 0$), so the function is defined everywhere on the real line.