Hour:
Name:
Date:
Pre-Calculus: Chapter 1 Review
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a) Explain why the number 5 is not in the domain of f.
1. Let f (x) =
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b) Explain why the number 25 is not in the domain of f.
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c) Find the domain of f.
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2. Find a function with the indicated domain.
a) (-00,-3)U(-3,6)U(6,00)
b) [-3,6)U(6,00)
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c) ( -3, 6 ) ( 6 , 00)
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3. Let f (x) = (x —1)2 — 4 .
a) Identify the domain and the range of f.
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b) Find f (2) .
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c) For what value(s) of x does f (x) = 5?
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d) Algebraically find the solution to the inequality (x —1)2 — 4 > 0.
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e) Accurately graph f (x) on the axes provided. Label the intercepts.
(THINK!: Does your graph confirm your answers to all of the
above problems?)
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f) Evaluate the difference quotient
f (x + h)— f (x)
— ( x2-
2 0G11)
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4. The graph of y = Ix is shown at right.
Sketch the graph of y = -21x -11 and describe the sequence of
transformations that occurs from the parent function to obtain the
graph you sketched.
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5. Given the graph of f at right, sketch the graph of y =if (-x)- 2,
2
describe the sequence of transformations that occurs from f to
obtain the graph you sketched.
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6. The domain of the one-to-one function f is (-00, 4)U (4,00 and its range is [2,00) . Find the
domain and range of each of the following functions.
Domain
Function
Range
f (x-3)
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f (3x)
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7. An isosceles triangle is inscribed in the curve f (x) = —2x2 + 24, as shown.
a) Express the area of the triangle as a function of x.
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b) Determine the domain of the area function.
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c) What is the maximum area of the triangle?
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8. Consider a square.
a) Write the area of a square as a function of the length of its diagonal.
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b) Express the length of the diagonal of a square as a function of its area.
c) What is the relationship between the two functions you just wrote? Does your answer need any
qualifications or stipulations? Are these justified in the context of the problem?
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9. The function f is either always increasing or always decreasing on the interval [2,25] . If f (7) = 8,
then how does the value f (12) help us decide whether f is increasing or decreasing? Be specific.
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10. Algebraically determine whether each function is even, odd, or neither.
b) g(x)=x(x+1)(x-1)
a) f (x) = x2 -2x +7
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11. Consider even functions.
a) What does it mean algebraically for a function to be even?
b) What does it mean graphically for a function to be even?
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c) Use your algebraic definition to show that f (x) = x4 -17 is even and g (x) = (x -2)2 is not.
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d) Confirm your findings in part (c) using the graphs of f and g. Explain how you did this.
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12. The graph of g passes through the points (4,9) , (a,11), and (-b, -c).
a) Name three additional points on the graph of g if g is odd.
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b) Name three additional points on the graph of g if g is even. C.
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c) Name three points that cannot be on the graph of g if g is one-to-one.
d) Name three points on the graph of g-1 if g is one-to-one.
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13. Let f (x) = 2-x'.
a) Find the function g such that g (x) = f (x) .
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b) Algebraically verify that f and g are inverses of each other.
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c) Graphically verify that f and g are inverses of each other. Explain how you did this.
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14. Refer to the graph of h at right. (As always, express your answers in interval notation, where
appropriate.)
a) Find h (3) . t(10-3)::. b) Find WO . hc
c) For what value(s) of x is h (x) = 0 ? h(‘) ()
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d) On what interval is h(x) O?
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e) On what interval is h increasing?
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State the domain of h.
State the range of h.
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What is the minimum value of h on the interval shown? hc)
What type of function is h (even, odd, or neither)?,,
Explain why h is a function.
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15. Given: f (10) = -7
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f is a linear function -7
g (x) = x2 + 2x-3
0
Find: (f -0-2)
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16. Answer always, sometimes, or never.
a) The graph of a relation 5DM.QA(VQS
passes the vertical line test.
b) The graph of a function SM3A&s
passes the horizontal line test.
c) A one-to-one function is NCLAief
even.
d) If a function has an inverse, then it is al 1AJOJJS
e) A one-to-one function is 0.1k)( U,
either always increasing or always decreasing.
f) The composition of an odd function and its inverse is
g) An odd function
one-to-one.
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odd.
SOME:NAILS has an inverse.
h) If f and g are functions, then ( f + g) is iLL(AJQ.,t
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even.
i) Quadratic functions are
j) Quadratic functions are
one-to-one.
k) Cubic functions are S. ColvlitYAIS
1) Cubic functions are
at
odd.
one-to-one.
m) (f og)(x) is
equal to g (f (x)) .
n) A function is 50,1Ael Q
its own inverse.
o) f (a) is (OA)
a function.
called a relative maximum if there exists an interval (x1 , x2 ) that
contains a and if x1 <x <x2 implies f (a) f (x).
17. Let h(x) = \l(f g)(x) , where f (x) = +1 and g (x) = 6x — 4 . Find the domain of h.
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18, The equation S (n) = 180(n —2) shows the sum of the measures of the interior angles of a polygon as
a function of the number of sides of the polygon (i.e., input is the number of sides, output is the sum
of the measures of the interior angles).
a) Find 5" (n)
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b) What are the inputs and outputs of the inverse function?
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c) what is the domain of the inverse function?
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19. Let AB = 15.
a) Find the value of a. (341
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2
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b) Write the slope-intercept equation of AB.
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c) Find the coordinates of D.
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Name
UNIT 1 Reasoning and Proof
9/26/10
Date
Form B
1. In the diagram below, inZAFB = 32' and CG 1 AD.
Find the measures of the indicated angles. Provide reasoning to support your answers.
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2. In the diagram below, if inG1 = mG2, andf ji g, Prove that rtiLl + inZ3 = 180'.
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Unit 1 I Reasoning and Proof
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3. If mZ1. = ma which lines must be parallel? Explain your reasoning.
4. In the diagram below, 49. 11 in and mL7 = mL9. Prove that mZ2 = mL9.
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Unit 1 I Reasoning and Proof
Assessment Master I use after Lesson 2