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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
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