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.
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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.
3. G (x) = 4 x ; find g ( - 1/2).
a. Plugging -1/2 in for x yields 4 -½ = 1/2.
Solve by factoring
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
Graph the function
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)
Polynomial or linear function.
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 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 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
With a bit of arithmetic work in subtracting fractions this becomes -
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.
d. Graph the function.
In part a, 8 months is 2/3 years and 1 year and 4 months is 4/3 years.
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.