Practice Final
1
Math 1030/1030I – Spring 2012
Practice Final
1. Decide if each of the following equations defines a function as indicated:
(i) Does the equation 4x − y 2 = 1 define y as a function of x?
x
= 2 define t as a function of x?
(ii) Does the equation
2t − 8
(iii) Does the equation 4x − y 2 = 1 define x as a function of y?
√
(iv) Does the equation 4 3t + 8u = 1 define t as a function of u?
2. Let f (x) be defined by
f (x) =

 2x + 5 if x ≤ 2

8−x
if x > 2
Find the following:
a) f (0)
b) f (2)
c) f (3)
3. Find the domain of the functions, and write the answer in interval notation:
x−3
t2 + 1
(a) f (x) = √
(b) g(t) =
4t
2x − 5
(c) h(x) = 3x3 +
(e) G(x) =
√
x2
+ 2x + 2
2
√
x2 − x − 6
4x + 1
x2 − 4
√
(f) H(x) = 3x − 12
(d) F (x) =
4. Let f and g be functions given by the formulas
f (x) = 3x2 − 1
functions:
(a) f (x) + g(x)
g(x) = 1 − 2x Find formulas for the following
(b) f (x) − g(x)
(c) 3f (x) + 2g(x)
(d) f (x)g(x)
5. Let
f (x) = 3x + 2,
g(x) = 4 − x2
Find the following:
(i) f (g(x))
6.
(ii) g(f (x))
(iii) f (f (x))
(iv) g(g(x))
(i) Let f (t) = 2t − 7. Find t such that f (t) = 5.
(ii) Let g(x) = x2 − x − 20. Find all values of x such that g(x) = 0.
(iii) Let h(x) = x3 − 3x. Find all x such that h(x) = x.
(iv) Let f (x) = 2x2 + 2x − 10 and g(x) = x2 + 3x + 10.
Find all x such that f (x) = g(x).
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Math 1030/1030I
7. The graph of f (x) is shown in the picture.
f(x)
4
3
2
1
4
3
2 1
1
2
1 2
3
5 6 x
4
3
Find the following:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Domain of f
Range of f
Intervals where f (x) is positive
Intervals where f (x) is negative
x-intercept(s)
y-intercept
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
Intervals where f is increasing
Intervals where f is constant
Intervals where f is decreasing
Points where f (x) is zero
f (1)
Solutions of f (x) = 4
Is f (x) one-to-one?
8. Find the inverses of the following functions, if possible:
(a) f (x) = 5 − 2x3
(b) g(t) =
t+1
3−t
(c) h(x) = 9x2 − 3
(d) f (x) = 2x + 7
9. Find the difference quotient for the functions:
(a) f (x) = 4x − 1
(b) g(x) = 2x2 − 3x + 1
10. For the quadratic function f (x) = 2x2 − 8x − 10
(a) Find the vertex.
(b) Find the x– and y– intercepts.
(c) Write the function in standard form.
(d) Sketch the graph, labeling the vertex and the intercepts.
(e) What is the parent function of f (x), and how is the graph of f (x) related
to that of its parent function?
11. For the quadratic function f (x) = −3x2 − 6x
(a) Find the vertex.
Practice Final
3
(b) Find the x– and y– intercepts.
(c) Write the function in standard form.
(d) Sketch the graph, labeling the vertex and the intercepts.
(e) What is the parent function of f (x), and how is the graph of f (x) related
to that of its parent function?
12. Find the standard form of the quadratic functions with the given vertex and
going through the given point:
(a) Vertex at (1, −2), goes through (0, 2).
(b) Vertex at (−2, 0), goes through (1, −2).
(c) Vertex at (0, 3), goes through (4, 0).
13. Solve the inequalities, and write the answer in interval notation:
(a) x2 + 2x − 24 > 0
(c) (x + 2)(x − 1)2 < 0
(b) 4x2 − 9 ≤ 0
(d) (x − 3)(x − 2)3 ≥ 0
14. For the polynomial function f (x) = x3 − 4x2 + x − 6
(a) Use the Rational Zero Theorem to list all the possible rational zeros of
f (x).
(b) Decide if x − 2 is a factor of f (x).
(c) Find all the zeros of f (x).
(d) Write f (x) in factored form.
(e) Sketch the graph of f (x), labeling all intercepts.
15. For the polynomial function f (x) = 2x4 − 7x3 + x2 + 7x − 3
(a) Use the Rational Zero Theorem to list all the possible rational zeros of
f (x).
(b) Use synthetic division to find at least two zeros of f (x).
(c) Find all the other zeros of f (x).
(d) Write f (x) in factored form.
16. For the rational function h(x) =
(a) Find its domain
(b) Find all holes
(c) Find all intercepts
(d) Find all asymptotes
3+x
x−2
(e) Find some extra points, if necessary
(f) Sketch an accurate graph of the
function, labeling all intercepts
and asymptotes
4
Math 1030/1030I
17. For the rational function h(x) =
3x + 6
x+1
(e) Find some extra points, if necessary.
(a) Find its domain.
(b) Find all holes.
(f) Sketch an accurate graph of the
function, labeling all intercepts
and asymptotes.
(c) Find all intercepts.
(d) Find all asymptotes.
18. Find the exact value of the following expressions (no calculator approximations).
(a) log 10113
(b) ln (ex+5 )
(c) e3 ln 6
(d) loga a2
1
16
1
(h) log √
10
(e) log3 9
(f) log9 3
(g) log4
(i) log6 12 + log6 3
(j) log2 32 − log2 8
(k) 2 ln e3 − 3 ln e5
(l) log2 45
19. For each of the following logarithmic functions,
• Describe in words how the graph
of the function is related to the
graph of y = ln x.
• Find ALL intercepts.
• Find ALL asymptotes.
(a) f (x) = log(x − 2)
• Draw the graph, labeling intercepts and asymptotes.
• Find the domain and range of the
function.
(b) f (x) = − ln(x + 1).
20. The graphs of f (x), g(x) and h(x) are shown in the picture.
10
8
6
4
2
1
3
20
2
2
3
4
15
10
5
5
4
1
1
5
2
6
2
2
1
1
2
1
8
3
f (x)
10
g(x)
h(x)
For each of the graphs:
(a) Find vertical and horizontal asymptotes (if any).
(b) Find x- and y-intercepts (if any).
(c) Match the graph with the following equations:
(i) y = ex − 1
(ii) y = ln(x − 2)
(iii) y =
2x − 10
x+5
10
15
20
Practice Final
5
21. Expand the following as a sum, difference, or multiple of simple logarithms.
4√ 2
x t
(x − 4)(3x + 2)3
√
(a) log2
(b) ln
4y 5
2x − 1
22. Condense the following into a single logarithm:
1
1
(b) log2 (x − 4) − [log2 (x + 4) + log2 (x − 1)]
(a) 4 log x + 2 log y − log z − log t
2
2
23. Solve the following equations for x. Give exact answers (no calculator approximations).
(a) 82x+1 = 4
(b) ex−3 = 2
(c) 5 − 23x+1 = 2
(d) 23x+1 = 5x−2
(e) ln(x − 5) + ln(x + 1) = ln(3 − 2x)
(f) log3 x + log3 (x − 2) = 1.
24. Convert to degree measure, and simplify the fraction. Give exact answers (no
decimal approximations).
π
2π
(a) θ = 30
(b) θ =
(c) θ =
15
5
25. Convert to radian measure, and round the answer to two decimal places.
(a) θ = 170◦
(b) θ = 35◦
(c) θ = −315◦
26. Use the arc length formula to find the degree measure of θ. Give answers with
two decimal places.
(a) r = 4 in, s = 5 in
(b) r = 10 ft, s = 2π ft
27. Find the angle θ (in radians) for each of the following pictures:
7 ft
6 in
3 ft
6 in
θ
θ
(a)
(b)
28. Find the exact value (no decimal approximations):
π
(a) sec
(b) sin 60◦
4
(c) cot 30◦
29. Find the reference angle θR for the given angle θ, and then find the exact value
of all trigonometric functions of θ.
π
(a) θ = 315◦
(b) θ = −
6
30. Compute the exact value of the six trigonometric functions of angle θ if the
given point lies on its terminal side (no decimal approximations).
(a) (12, −5)
(b) (−2, 7)
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Math 1030/1030I
31. Find all trig functions of θ given the following information:
5
3
(a) sin θ = , cos θ < 0
(b) cos θ = − , tan θ > 0
8
5
32. Find period, amplitude, and shifts for each of the following, then sketch one full
period:
π
(a) f (x) = 2 cos(4x)
(b) f (x) = − cos x +
2
33. Find period and shifts for each of the following, then sketch one full period:
π
(a) f (x) = tan(2x)
(b) f (x) = − tan x −
4
34. Find the exact value:
8
−1
−
(a) cos tan
5
−1
(b) sin cos
5
13
35. Re-write the expression using the given trig substitution and simplify:
√
√
(a) 25 − x2 , x = 5 sin θ
(b) 16 + x2 , x = 4 tan θ.
36. Verify the identities:
(a) csc θ − sin θ = cos θ cot θ
(b) sin θ tan θ = sec θ − cos θ
37. Solve the equation 2 cos θ + 1 = 0 for all values of θ in the interval [0, 2π).
38. Find all solutions of the equation 3 tan2 θ − 1 = 0.
39. Use sum/difference identities to find the exact value (no decimal approximations):
(a) cos(75◦ )
(b) tan(15◦ )
40. Use sum/difference identities to find the exact value (no decimal approxima1 1
5
1 1
1
= + ].
tions) [Hint: = − , and
12
4 6
12
4 6
π
5π
(a) sin
(b) cos
12
12
41. Suppose that cos θ = 0.1. Find the exact value of cos(2θ).
42. Suppose that cos θ = −
sin(2θ).
5
and θ is in Quadrant II. Find the exact value of
13
43. Find all solutions in the interval [0, 2π):
(a) cos(2x) + sin2 x = 1
44. Solve the equation in the interval 0◦ ≤ θ ≤ 360◦ :
π
π
sin x −
+ cos x −
= 1.
4
4
(b) sin(2x) + cos x = 0.
Practice Final
7
45. Solve the triangle:
B
5
A
75
75
50
35
C
46. Solve the triangle:
B
12
7
A
10
C