Let's tackle both problems from your image step by step.

Problem 29:

We are given: $H(x) = \frac{\sqrt{1 + \sqrt{x}}}{\sqrt{2 + \sqrt{x}}}$ and $g(x) = \sqrt{x}.$

(a) Find a function $f$ such that $h = f \circ g$.

We want to express $H(x)$ in the form of $f(g(x))$, meaning we need to find an $f(x)$ such that: $H(x) = f(g(x)) = f(\sqrt{x}).$ By inspection, comparing $H(x)$ and $g(x)$, we see that $H(x)$ involves nested square roots. If we substitute $g(x) = \sqrt{x}$ directly into $H(x)$, we need to match it to: $f(\sqrt{x}) = \frac{\sqrt{1 + \sqrt{x}}}{\sqrt{2 + \sqrt{x}}}.$ Thus, the function $f(x)$ that we need would simply be: $f(x) = \frac{\sqrt{1 + x}}{\sqrt{2 + x}}.$

(b) Find a function $f$ such that $h = g \circ f$.

Now we need to find an $f(x)$ such that $h(x) = g(f(x)) = \sqrt{f(x)}$.

Since $H(x)$ is expressed as: $H(x) = \frac{\sqrt{1 + \sqrt{x}}}{\sqrt{2 + \sqrt{x}}},$ we need $f(x)$ to satisfy: $\sqrt{f(x)} = \frac{\sqrt{1 + \sqrt{x}}}{\sqrt{2 + \sqrt{x}}}.$ To solve for $f(x)$, square both sides: $f(x) = \left( \frac{\sqrt{1 + \sqrt{x}}}{\sqrt{2 + \sqrt{x}}} \right)^2 = \frac{1 + \sqrt{x}}{2 + \sqrt{x}}.$

So, the function $f(x)$ is: $f(x) = \frac{1 + \sqrt{x}}{2 + \sqrt{x}}.$

Problem 47:

We are given: $f(c) = \frac{c}{91.44}$ where $f(c)$ gives the exact number of yards in $c$ centimeters.

(a) Find a formula for $f^{-1}(y)$.

To find the inverse of $f(c)$, we need to solve for $c$ in terms of $y$ (the number of yards). Start with: $y = \frac{c}{91.44}.$ Multiply both sides by 91.44 to solve for $c$: $c = 91.44y.$ Thus, the inverse function is: $f^{-1}(y) = 91.44y.$

(b) What is the meaning of $f^{-1}(y)$?

The function $f^{-1}(y)$ gives the number of centimeters corresponding to $y$ yards. In other words, if you know the number of yards $y$, $f^{-1}(y)$ converts that value back into centimeters.

Would you like any further details or explanations?

Here are some follow-up questions:

1. How do you generally find the inverse of a function?
2. Why does the function composition involve substitution?
3. Can all functions be composed in this way (like in 29(a) and 29(b))?
4. Why is squaring necessary in part 29(b)?
5. How do unit conversions relate to function inverses, as in problem 47?

Tip: When finding the inverse of a function, always check if the inverse is valid by substituting it back into the original function to see if you retrieve the identity function (i.e., $f(f^{-1}(x)) = x$).