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) = (1)/(2)x mg (amount in bloodstream from oral dose x) • Amount absorbed into infection site: g(x) = (4x)/(x+4) mg (amount at infection site from x mg in bloodstream) • Number of surviving bacteria: f(x) = (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) = (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((1)/(2)x) = (4(1)/(2)x)((1)/(2)x)+4 = (2x)/(x)2+4 = (2x)/(x+8)2 = (4x)/(x+8) Let A_oral(x) = (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((4x)/(x+8)) = (3200)/(8 + (4x)x+8)^2 = (3200)/(8 + 16x^2)(x+8)^2 = (3200)/(8(x+8)^2 + 16x^2)(x+8)^2 = (3200(x+8)^2)/(8(x^2+16x+64) + 16x^2) = (3200(x+8)^2)/(8x^2+128x+512 + 16x^2) = (3200(x+8)^2)/(24x^2+128x+512) Factor out 8 from the denominator: = (3200(x+8)^2)/(8(3x^2+16x+64)) = (400(x+8)^2)/(3x^2+16x+64) The function relating oral dosage to surviving bacteria is: P_oral(x) = (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) = (4x)/(x+4). Let A_inject(x) = (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((4x)/(x+4)) = (3200)/(8 + (4x)x+4)^2 = (3200)/(8 + 16x^2)(x+4)^2 = (3200)/(8(x+4)^2 + 16x^2)(x+4)^2 = (3200(x+4)^2)/(8(x^2+8x+16) + 16x^2) = (3200(x+4)^2)/(8x^2+64x+128 + 16x^2) = (3200(x+4)^2)/(24x^2+64x+128) Factor out 8 from the denominator: = (3200(x+4)^2)/(8(3x^2+8x+16)) = (400(x+4)^2)/(3x^2+8x+16) The function relating injected dosage to surviving bacteria is: P_inject(x) = (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. P_oral(3) =