Chapter 1 Test
1. Answer true or false. The sine is an odd function.
2. Answer true or false. The cosine is an odd function.
3. Answer true or false. The power function with an odd exponent is an even function.
4. Answer true or false. There exists a function that is both even and odd.
5. Answer true or false. The natural exponential function is invertible.
6. Answer true or false. When the period of f ( x) is p, the period of f ( x / 3) is p/ 3
7. Answer true or false. The exponential function y= 2x has no x-intercept.
8. Answer true or false. The function's graph cannot cross its asymptotes.
9. Answer true or false. log 3 ( x y )= log 3 ( x ) log 3 ( y ).
10. Answer true or false. The number log 2 (32 ) is equal to 5.
11. A clipping from the graph of f ( t )=t p, where p is an integer, is shown.
Based on this clipping
(a) the exponent p is odd and positive,
(b) the exponent p is even and positive,
(c) the exponent p is odd and negative,
(d) the exponent p is even and negative,
(e) we cannot determine the parity and/or polarity of p.
12. Suppose that f ( t )= 3 t 4 +29 t 2 2 t +1 t 4 13 t 2 + 36 . Select the most accurate of the statements below.
(a) The graph of f has vertical asymptote(s).
(b) The graph of f has a horizontal asymptote.
(c) The graph of f has vertical asymptote(s) and a horizontal asymptote.
(d) The graph of f has no horizontal or vertical asymptotes.
13. The wavelength of sin ( πx ) is …
(a) 2
(b) 1
(c) 2 π
(d) π
(e) none of the above
14. Determine which of the following equations are correct.
(a) ( ln ( x + 2 ) )5 =5 ln ( x+ 2)
(b) 2 ln ( x )+ ln ( 7 )= ln (7 x 2)
(c) both equations (a) and (b) are correct
(d) neither equation is correct
15. Use the parity, polarity, and magnitude that you see in the exponent to compare and contrast the way that f ( x )= x 3 /5 and g ( x )= x 4 /5 change when x moves away from zero.
16. Use your understanding of parity to determine the behavior of the polynomial f ( t )= 2 t 7 +3 t 6 5 t 3 +9 for large |t|
17. Sketch a rough graph of the polynomial g ( t )=( t 1 )2 ( 3 t ) (t +1 ) based on its behavior near its roots, the sign of the leading term, and the degree of the polynomial.
18. Sketch a rough graph of the polynomial h ( t )=( t + 2 )2 ( t +1 ) ( 1 t )3 based on its behavior near its roots, the sign of the leading term, and the degree of the polynomial.
19. Determine/design a polynomial function whose graph could be the one shown below:
20. Determine the behavior of the function f ( t )= 2 t 5 + 3
21. Determine the behavior of the function g ( t )=
22. Sketch a graph of the function f ( t )= 2
2 3 t 2 + 1 .
23. Sketch a graph of the function g ( t )= t + 4 1 t 2 .
for large |t|.
+ 5 for large |t|
24. For f ( t )= 6 t 7 2 +t , find (a) equations of all vertical asymptotes (if any), (b) an equation of the horizontal asymptote (if any), (c) the slope of the oblique asymptote (if any)
25. For g ( t )= 12 t 3 +5 t +1 4 t 2 +1 , find (a) equations of all vertical asymptotes (if any), (b) an equation of the horizontal asymptote (if any), (c) the slope of the oblique asymptote (if any)
26. For h ( t )= 8 t 5 +3 t 6 9 t 4 , find (a) equations of all vertical asymptotes (if any), (b) an equation of the horizontal asymptote (if any), (c) the slope of the oblique asymptote (if any)
27. For f ( t )=2 √ t +3 and g ( t )= t 2 +3 t 1 determine a formula for (a) ( f ∘ g) (t ), and (b) ( g ∘ f ) (t )
28. Sketch the graph of the piecewise defined function y ( t ) ={ t ∧ if t ← 2 t 2 1 ∧if t ≥ 2 .
29. Sketch the graph of y= 2 cos ( 4 x 3 )+1.
30. For f ( t )=2 sin ( 3 t )+2, sketch y= f (t ) and y=1 / f (t ) on the same axes.
31. Determine the wavelength, angular frequency, and speed of the traveling wave cos (8 x 9 t ).
32. Use the properties of exponents to rewrite A ( t )=7 ( 0.6 )3 t as an exponential function whose base is larger than one.
33. Suppose that f ( t )=2.2 t and g ( t )=2.4 t. Which function decreases faster as t becomes more and more positive?
34. Evaluate log 3 ( 1 27 ) without using a calculator.
35. Use a calculator to determine log 5 ( 7 π ).
36. Solve the equation 3 x 2 =74 x
37. Without a calculator, determine which of log 2 e and ln 2 is larger.
38. Evaluate arctan (cos ( 3 π ) ) without a calculator.
39. Rewrite cot ( sin 1 (√ x)) without trigonometric functions.
40. Find a formula for f 1 for f ( t )=3 4 2t .
Answers
1. true
2. false
3. false
4. true
5. true
6. false
7. true
8. false
9. false
10. true
11. (d)
12. (c)
13. (a)
14. (b)
15. f ( x) and g( x ) both move away from y= 0, curving downward on the right; f ( x) has a range of all real numbers, but g( x ) is non-negative
16. As t → ∞ , f ( t ) → ∞; as t → ∞ , f (t ) → ∞
17.
20. As |t|→ ∞ , f (t ) → ∞
21. As |t|→ ∞ , g(t )→ 3
22.
23.
24. (a) t = 2, (b) y= 6, (c) none
25. (a) none, (b) none, (c) 3
26. (a) t = 3, t =0, t =3, (b) y= 0, (c) none
27. (a) 2 √ t 2 +3 t 1 + 3, (b) ( 2 √t +3 )2 +3 2 √t +2
28.
31. λ= π 4 , ω=9, s= 9 8
32. A ( t )=7 ( 10 6 ) 3 t
33. g ( t )=2.4 t
34. 3
35. log 5 ( 7 π ) ≈ 1.9203
36. x=0, x= 4 ln (7 ) ln ( 3 )
37. log 2 e
38. π 4
39. √ 1 x x
40. log 2 ( log 4 ( 3 t ) )