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:a.Evaluate the given expression b.Find the domain of the function. c.Find the range. [Hint: Use a graphing calculator]

2. 2x

f (x)x 1

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.

Solve by factoring

4. 5 3 1

2 2 22x 4x 6x 0

03x2xx2 2

2

2

4

2

1

Factor out the common factor 2 x ½ .

1

222x x 2x 3 0

1

22x x 1 x 3 0 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.

f (x) 3x

6.

f (x) x 3 3

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

Graph the function

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

7.

f (x) x 5

8.

f (x) x 2

Polynomial or linear function.

9.

3xx47

3x2x)x(f

10.

f (x) x 2 x1

2

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

For 11-14 each function find and simplifyAssume h 0.

2 h 10x 5hf (x h) f (x) 10xh 5h10x 5h

h h h

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.

f (x h) f (x)

h

12.

f (x) 7x 2 3x 2

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

2 h 14x 7h 3f (x h) f (x) 14xh 7h 3h14x 7h 3

h h h

Step 4.

13.

f (x) x 33 3 2 2 3[h int :use (x h) x 3x h 3xh h ]

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

2 22 2 32 2

h 3x 3xh hf (x h) f (x) 3x h 3xh h3x 3xh h

h h h

Step 4.

14.

f (x) 2

x

Step 1. hx

2)hx(f

Step 2.

f (x) 2

x

Step 3. x

2

hx

2)x(f)hx(f

With a bit of arithmetic work in subtracting fractions this becomes -

2h

x(x h)

f (x h) f (x)

h

Step 4. We are dividing step 3 by h or multiplying by 1/h.

2h 1 2

x(x h) h x(x 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 incomein 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 $10000d. Graph the function.

0.10x if 0 x 5000f (x)

500 0.30(x 5000) if x 5000

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.

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 incomein excess of $5000.

d. Graph the function.

5000xif)5000x(30.0500

5000x0if10.0)x(f

17. The usual estimate that each human-year corresponds to 7 dog-yearsis not very accurate for young dogs, since they quickly reach adulthood. a. Find the number of dog years corresponding to the following amounts ofhuman 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 yearfor the first 2 years , and then 4 dog years per human years for each year thereafter: 10.5x if 0 x 2

f (x)21 4(x 2) if x 2

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-yearsis not very accurate for young dogs, since they quickly reach adulthood. a. Find the number of dog years corresponding to the following amounts ofhuman 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 yearfor the first 2 years , and then 4 dog years per human years for each year thereafter: 10.5x if 0 x 2

f (x)21 4(x 2) if x 2

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 mousewill divide into two cells every 12 ours, so that after x days the number of leukemiccells 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 itc.survive beyond day 15?