Derive the function that relates oral dosage to the number of surviving bacteria using composition of functions.

Here's how to solve the problem using function composition:

The given functions are:

• Bioavailability (oral): $h(x) = \frac{1}{2}x$ mg (amount in bloodstream from oral dose $x$)
• Amount absorbed into infection site: $g(x) = \frac{4x}{x+4}$ mg (amount at infection site from $x$ mg in bloodstream)
• Number of surviving bacteria: $f(x) = \frac{3200}{8+x^2}$ CFU (surviving bacteria from $x$ mg at infection site)

a) Derive the function that relates oral dosage to the number of surviving bacteria using composition of functions.

Step 1: Determine the amount of antibiotic in the bloodstream from an oral dose $x$.
This is given by $h(x) = \frac{1}{2}x$.

Step 2: Determine the amount of antibiotic absorbed into the infection site from the amount in the bloodstream.
This is $g(h(x))$. Substitute $h(x)$ into $g(x)$:

$$g(h(x)) = g\left(\frac{1}{2}x\right) = \frac{4\left(\frac{1}{2}x\right)}{\left(\frac{1}{2}x\right)+4} = \frac{2x}{\frac{x}{2}+4} = \frac{2x}{\frac{x+8}{2}} = \frac{4x}{x+8}$$

Let $A_{oral}(x) = \frac{4x}{x+8}$ be the amount at the infection site for an oral dose $x$.

Step 3: Determine the number of surviving bacteria from the amount at the infection site.
This is $f(A_{oral}(x))$. Substitute $A_{oral}(x)$ into $f(x)$:

$$f(A_{oral}(x)) = f\left(\frac{4x}{x+8}\right) = \frac{3200}{8 + \left(\frac{4x}{x+8}\right)^2}$$ $$= \frac{3200}{8 + \frac{16x^2}{(x+8)^2}} = \frac{3200}{\frac{8(x+8)^2 + 16x^2}{(x+8)^2}}$$ $$= \frac{3200(x+8)^2}{8(x^2+16x+64) + 16x^2} = \frac{3200(x+8)^2}{8x^2+128x+512 + 16x^2}$$ $$= \frac{3200(x+8)^2}{24x^2+128x+512}$$

Factor out 8 from the denominator:

$$= \frac{3200(x+8)^2}{8(3x^2+16x+64)} = \frac{400(x+8)^2}{3x^2+16x+64}$$

The function relating oral dosage to surviving bacteria is:

$$\boxed{P_{oral}(x) = \frac{400(x+8)^2}{3x^2+16x+64}}$$

b) Suppose the antibiotics is instead administered by injection. Derive the function that relates dosage to the number of surviving bacteria using composition of functions.

Step 1: If administered by injection, the entire dose $x$ mg is directly in the bloodstream. So, the amount in the bloodstream is simply $x$.

Step 2: Determine the amount of antibiotic absorbed into the infection site from the injected dose $x$.
This is $g(x) = \frac{4x}{x+4}$. Let $A_{inject}(x) = \frac{4x}{x+4}$ be the amount at the infection site for an injected dose $x$.

Step 3: Determine the number of surviving bacteria from the amount at the infection site.
This is $f(A_{inject}(x))$. Substitute $A_{inject}(x)$ into $f(x)$:

$$f(A_{inject}(x)) = f\left(\frac{4x}{x+4}\right) = \frac{3200}{8 + \left(\frac{4x}{x+4}\right)^2}$$ $$= \frac{3200}{8 + \frac{16x^2}{(x+4)^2}} = \frac{3200}{\frac{8(x+4)^2 + 16x^2}{(x+4)^2}}$$ $$= \frac{3200(x+4)^2}{8(x^2+8x+16) + 16x^2} = \frac{3200(x+4)^2}{8x^2+64x+128 + 16x^2}$$ $$= \frac{3200(x+4)^2}{24x^2+64x+128}$$

Factor out 8 from the denominator:

$$= \frac{3200(x+4)^2}{8(3x^2+8x+16)} = \frac{400(x+4)^2}{3x^2+8x+16}$$

The function relating injected dosage to surviving bacteria is:

$$\boxed{P_{inject}(x) = \frac{400(x+4)^2}{3x^2+8x+16}}$$

c) If someone has to take 3 mg of antibiotics, which treatment is the best?

To find the best treatment, we need to calculate the number of surviving bacteria for both oral and injection treatments with a 3 mg dose ($x=3$) and choose the one with the lowest number of surviving bacteria.

Step 1: Calculate surviving bacteria for oral treatment with $x=3$ mg.

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