Pg. 127/138 Homework

• This is the HW from after the test.

• #25 (3, -4) r = 4 #26 (1, -3) r = 7• #27 (2, -3) r = #28 (7, 4) r =

• #42 [-2, -1)U(-1, ∞) #43 (∞ , -1]U[4, ∞ )• #48 No real solutions #88 y – axis • #89 x – axis #90 no symmetry• #91 origin #92 y – axis, x – axis and origin• #93 origin

2 19

Pg. 136/150 Homework• Pg. 136 #10 – 34 even

Pg. 150 #45 – 49 all

• #9 True #27 Circle r = 2, C(0, 2)• #11 True #29 f -1(x) = x2 + 2• #13 f(g(x) = g(f(x) = x D:[0, ∞), R:[2, ∞)• #15 f(g(x) = g(f(x) = x #31 f -1(x) = ½(x) – (5/2)• #17 No, it fails the HLT #33 f -1(x) = (2x + 3)/(x – 1)• #19 No, it fails the HLT #35 f -1(x) = x2 + 2• #21 Yes, it passes the HLT #42 (a) 450 – 15x = rent • #23 No (b) 1900+20x =tenants• #25 Where they meet #43 [0, 30)

on the line y = x #44 x = 0, rent = $450

2.7 Inverse Functions

Pg. 150• A large apartment rental

company has 2500 units available, and 1900 are currently rented at an average of $450/mo. A market survey indicates that each $15 decrease in average monthly rent will result in 20 new tenants.

#42 – 44 • #42 – Money and people stick together.

That’s why there are two equations:450 – 15x = rent1900 + 20x = tenants

• #43 – Revenue is the number of tenants times the cost of rent, or those two equations multiplied. So, R = (450 – 15x)(1900 + 20x). Set each piece equal to zero and you know you’re boundaries. x = 0 and x = 30, so [0, 30) because you don’t want R = 0.

• #44 – Graph R in your calculator and find the maximum. It will occur when x = 0 and R = 450.

2.7 Inverse Functions

Inverse Relations• The point (a, b) is in the

relation R if, and only if, (b, a) is in the relation R-1.

• Graphically, an inverse is a reflection of the original graph over the line y = x.

Inverse Functions• In order for an inverse

function to exist, first you must be dealing with a function and that function must pass the VLT and the HLT.

• Functions and Inverse Functions can be composed together to prove they are inverses of each other. Their result will always be x.

2.7 Inverse Functions

Examples• Find the inverse of

y = ½ x – 3 algebraically.

• Graph the original equation and the inverse along with the line y = x to show it has the proper symmetry.

• Find the inverse of f(x) = -x3 algebraically.

• Graph the original equation and the inverse along with the line y = x to show it has the proper symmetry.

2.7 Inverse Functions

Inverse Functions• Show that f(x) =

will have an inverse function. – Find the inverse function and

state its domain and range. – Prove that the two are

actually inverses.

• Show that g(x) = will have an inverse function. – Find the inverse function and

state its domain and range. – Prove that the two are

actually inverses.

• Show that h(x) = x3 – 5xwill have an inverse function.

4x 3 4

2

x

x

3.1 Graphs of Polynomial Functions

Definition • A polynomial function is

one that can be written in the form:

where n is a nonnegative integer and the coefficients are real numbers. If the leading coefficient is not zero, then n is the degree of the polynomial.

State whether the following are polynomials. If so, state the degree.

11 1 0...n n

n nf x a x a x a x a

5 9f x x

22 9 3f x x x

6 4 3 23 7 2 5 9f x x x x x

2 9f x x

3.1 Graphs of Polynomial Functions

End Behavior• End behavior is determined

by the degree and the leading coefficient.

• Create Chart.

Number of “Bumps”• The number of “bumps” a

graph may have is no more than one less than the degree.

• The number of zeros a graph may have is no more than the number of the degree.

22 9 3f x x x

5 4 3 23 7 2 5 9f x x x x x