Section 1.4
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Transcript of Section 1.4
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?