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Section 1.4

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Section 1.4

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.

- 2-3 For each function:
- Evaluate the given expression
- Find the domain of the function.
- Find the range. [Hint: Use a graphing calculator]

2.

- 2-3 For each function:
- Evaluate the given expression
- 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.

- 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.

Solve by factoring

4.

Factor out the common factor 2 x ½ .

So x = 0. x = 1 and x = -3

You can also graph this function on your calculator and find the x-intercepts – zeros.

Graph the function

5.

Graph the function

6.

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

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.

8.

Polynomial or linear function.

9.

10.

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

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.

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.

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.

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.

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.

- 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.
- Calculate the tax on an income of $3000.
- Calculate the tax on an income of $5000.
- Calculate the tax in an income of $10000
- Graph the function.

- 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.

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.

- 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.
- Find the number of dog years corresponding to the following amounts of
- human time: 8 months, 1 year and 4 months, 4 years, 10 years.
- b. Graph the function
- The following function expresses 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:

In part a, 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.
- Find the number of dog years corresponding to the following amounts of
- human time: 8 months, 1 year and 4 months, 4 years, 10 years.
- b. Graph the function
- The following function expresses 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:

17. Conti

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.

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