Use the given function and compute the first six iterates of each initial input $x_0$. In cases in which a calculator answer contains four or more decimal places, round the final answer to three decimal places. (However, during the calculations, work with all of the decimal places that your calculator affords.)$g(x)=\frac{1}{4}x+3$(a) $x_0=3$(b) $x_0=4$(c) $x_0=5$

Step by Step Solution: Step 1. Substitute the initial values of x into the function g(x),
For a: g(3) = (1/4)*3 + 3 = 3.75
For b: g(4) = (1/4)*4 + 3 = 4
For c: g(5) = (1/4)*5 + 3 = 4.25 Step 2. Use the result obtained in step 1 to calculate the next iterate by repeatedly applying the function,
For a: g(3.75) = 3.938
For b: g(4) = 4
For c: g(4.25) = 4.563 Step 3. Repeat step 2 until you get the desired number of iterates,
For a: g(3.938) = 4.098
g(4.098) = 4.174
g(4.174) = 4.218
g(4.218) = 4.238
g(4.238) = 4.248
For b: g(4) = 4
g(4) = 4
g(4) = 4
g(4) = 4
g(4) = 4
For c: g(4.563) = 4.698
g(4.698) = 4.924
g(4.924) = 5.155
g(5.155) = 5.311
g(5.311) = 5.399 Final Answer: For a: x6 = 4.248
For b: x6 = 4
For c: x6 = 5.399