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Section 1.3 1. Find the Domain and Range of the function below. The domain is x -4. The graph does not cross a vertical line at x = -4. it has a vertical asymptote at x = - 4. The range is y 0. The graph does not cross the x axis which has an equation of y = 0. it has a horizontal asymptote at y = 0.

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2-3 For each function: a.Evaluate the given expression at the given x value. b.Find the domain of the function. c.Find the range. [Hint: Use a graphing calculator] 2. At x = - 1

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2-3 For each function: a.Evaluate the given expression b.Find the domain of the function. [Hint: Use a graphing calculator] 3. g (x) = 4 x ; find g ( - 1/2). a. Plugging -1/2 in for x yields 4 -½ = 1/2. b.Graph the function and the table will show that all x work for the domain. OR Note that the function does not have division or even roots so all real numbers work.

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4. Solve Factor out the common factor 2 x ½. So x = 0. x = 1 and x = - 3 BUT it contains a square rood so – 3 will not work. You can also graph this function on your calculator and find the x- intercepts – zeros.

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Graph the function 5.

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6. It is the absolute value function shifted 3 down and 3 to the right. Graph the function

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7-10 Identify each function as a polynomial, a rational function. an exponential function, a piecewise linear function, or none of these. (Don’t graph them, just identify their types) 7.

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8. Polynomial or linear function.

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9.

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10. It is not a polynomial function because one of the exponents is not an integer.

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For 11-14 each function find and simplify Assume h 0. 11. f (x) = 5x 2. Step 1. f(x + h) = 5 (x + h) 2 = 5x 2 + 10 xh + 5h 2 Step 2. f(x) = 5x 2 Step 3. f(x + h) – f (x) = 10 xh + 5h 2 Step 4.

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12. Step 1. f(x + h) = 7 (x + h) 2 – 3 (x + h) + 2 = 7x 2 + 14 xh + 7h 2 -3x – 3h + 2 Step 2. f(x) = 7x 2 – 3x + 2 Step 3. f(x + h) – f (x) = 14 xh + 7h 2 – 3h Step 4.

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13. Step 1. f(x + h) = (x + h) 3 = x 3 + 3x 2 h + 3xh 2 + h 3 Step 2. f(x) = x 3 Step 3. f(x + h) – f (x) = 3x 2 h + 3xh 2 + h 3 Step 4.

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14. Step 1. Step 2. Step 3. With a bit of arithmetic work in subtracting fractions this becomes - Step 4.We are dividing step 3 by h or multiplying by 1/h.

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15. Social Science: World Population The world population (in millions) since the year 1700 is approximated by the exponential function p (x) = 522 (1.0053) x where x is the number of years since 1700 (for 0 ≤ x ≤ 200) Using a calculator, estimate the world population in the year 1750.

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16. Economics: Income Tax The following function expresses an income tax that is 10% for incomes below $5000, and otherwise is $500 plus 30% of income in excess of $5000. a.Calculate the tax on an income of $3000. b.Calculate the tax on an income of $5000. c.Calculate the tax in an income of $10000 d.Graph the function. b.For x = 5000 use f(x) = 500 + 0.30(x – 5000) f (x) = 500 + 0.30(5000 – 5000) = $500 c. For x = 10000 use f(x) = 500 + 0.30(x – 5000) f (x) = 500 + 0.30(10000 – 5000) = 500 + 1500 = $2000.

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16. Economics: Income Tax The following function expresses an income tax that is 10% for incomes below $5000, and otherwise is $500 plus 30% of income in excess of $5000. d. Graph the function.

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In parts a & b, 8 months is 2/3 years and 1 year and 4 months is 4/3 years. 17. The usual estimate that each human-year corresponds to 7 dog-years is not very accurate for young dogs, since they quickly reach adulthood. A more accurate estimate is the following function expressing dog years as 10.5 dog years per human year for the first 2 years, and then 4 dog years per human years for each year thereafter:

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c. d.

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17. Conti e.

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18. BONUS HOMEWORK! Business: Insurance Reserves: An insurance company keeps reserves (money to pay claims) of R (v) = 2v 0.3, where v is the value of all if its policies, and the value of it’s policies is predicted to be v (t) = 60 + 3t, where t is the number of years from now. (Both R and v are in the millions of dollars.) Express the reserves R as a function of t, and evaluate the function at t=10.

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19. Biomedical: Cell Growth One leukemic cell in an otherwise healthy mouse will divide into two cells every 12 hours, so that after x days the number of leukemic cells will be f (x) = 4 x. a.Find the appropriate number of leukemic cells after 10 days. b.If the mouse will die when its body has a billion leukemic cells, will it survive beyond day 15?

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Rational Expressions, Vertical Asymptotes, and Holes.

Rational Expressions, Vertical Asymptotes, and Holes.

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