BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH0 Review Sheet. Given the functions f and g described by the graphs below: y = f(x) y = g(x) (a) Find: i. The domain of f ii. The range of f iii. An interval on which f is increasing iv. An interval on which f is decreasing v. The domain of g vi. The range of g vii. An interval on which g is one-to-one (b) Evaluate the following, if they exist: i. g() ii. (f + g)() iii. (f g)() ( ) g iv. ( ) f v. (f g) () vi. (g g)( 5) vii. (f f) (). Let f(x) = x + x + and g(x) = x x. (a) Find the domains of f and g. Give your answer using interval notation.
(b) Evaluate, if defined: f (g(0)); g (f(0)); (f g) (0). Given the graph of y = f(x), answer the following questions. (a) Find the domain of f (b) Find the range of f (c) Over which intervals is f increasing? (d) Over which intervals is f decreasing? (e) Find f( ) and f() (f) Find all solutions to the equation f(x) = (g) Find the zeros of the function. (h) Does f have an inverse function? Explain.. For each of the functions f given below: A. f(x) = x B. f(x) = e x C. f(x) = log x + ( x) (a) Find the inverse function f. (b) Verify that f ( f (x) ) = f (f(x)) = x (c) Sketch a graph of y = f(x) and y = f (x) on the same set of coordinates. 5. Consider the functions: f(x) = e x and g(x) = lnx. Are f and g a pair of inverse functions? Justify your answer. 6. For each pair of functions f and g given below find f g and g f. (a) f(x) = x x + 5; g(x) = 5 x. (b) f(x) = x x 5 ; g(x) = 5x x (c) f(x) = x ; g(x) = x + 5 7. The graph of a parabola y = f(x) has axis of symmetry x =, vertex (,5), and f(0) =. (a) Write the equation of the parabola in standard form. (b) State the domain and the range of f. Page
(c) Sketch a graph of y = f(x). 8. For each of the the following polynomials p(x): A. p(x) = x x + B. p(x) = x + x x 6 C. p(x) = x + 7x + 6x x (a) List all possible rational roots of p(x), according to the Rational Zeros Theorem. (b) Factor p(x) completely. (c) Find all roots of the equation p(x) = 0. (d) Determine the end behavior of the graph of y = p(x). (e) Determine the y intercept of the graph of y = p(x) (f) Determine the x intercepts of the graph y = p(x) (g) Determine the local behavior of y = p(x) near the x-intercepts. (h) Use the above information to sketch a graph of y = p(x). 9. Find the remainder of the division of x 0x 5 + 60x + when divided by x. 0. For each of the following rational functions f A. f(x) = x + x + x x B. f(x) = x + x x x E. f(x) = x x + (a) Factor numerator and denominator and simplify if possible. C. f(x) = x 9 x x (b) Find the x and y intercepts of the graph of y = f(x) if they exist. (c) Find any vertical or horizontal asymptotes. (d) Determine how the sign of f(x) changes. (e) Use the above information to sketch a graph of y = f(x). D. f(x) = x x + x. Solve the following inequalities. Express your answer using interval notation. A. x + x 7x x + 6 0 B. x + x > C. x x + x 6x + 9x 0. Evaluate the following expressions. Give exact values whenever possible: (a) log 6 (b) log 9 (c) log b x y, given that log b x = and log b y = 6 (d) e x y given that e x = and e y = ( ) x (e) log a given that log y a (x) = and log a (y) = (f) lne (g) log 000 (h) log 7, ( rounded to the nearest hundredth (i) sin sin ) 6 ( (j) cos cos ) (k) cos ( sin ( ) ) Page
(l) sin(a + b), if sina =, cos b = 5 and 0 < a,b <. Find θ if (a) cos θ =, and < θ <. (b) sinθ =, and < θ <. (c) sinθ =, and < θ < Q. Solve the following equations: (a) log x log (x ) = (b) 7 x+ = 9 (c) sin x =, where x is in the interval [0,) (d) cos x + cos x + = 0, where x is in the interval [0,) 5. Verify the following identities: (a) tan x + = sec x (b) csc x sinx = cot xcos x (c) csc x cos xcot x = sin x (d) cos x sin x = tan x + tan x 6. For each of the following functions A. f(x) = sin(x ) B. f(x) = cos(x + ) C. f(x) = sin(x ) D. f(x) = ( x cos ) (a) Find the period of this function. (b) Find the amplitude of the graph of y = f(x) (c) Find the phase shift of the graph of y = f(x) (d) Sketch two complete cycles of the graph of y = f(x) The answers. (a) i. [,), ii. [,5), iii. [,), iv. [,], v. [ 5,5], vi. (,5], vii. [ 5,) or [,5] (b) i. 0, ii., iii. undefined, iv. indeterminate, v., vi, vii. (a) Domain of f is (, ), domain of g is (,) (b) f (g(0)) = ; g (f(0)) is undefined; (f g)(0) =. A. ( 7,] B. [,5] C. ( 7, ] and [0,] D. [,0] and [,] E. f( ) = 5; f() = F. { 5,, } G. 6, H. No. It is not one-to-one.. For the graphs see Figure. A. f (x) = x x B. f (x) = lnx + C. f (x) = x 5. They are not a pair of inverse functions: f is not one-to-one and thus it doesn t have an inverse function. Page
y = f(x) y = x y = f(x) y = x y = f (x) y = f(x) y = x y = f (x) y = f (x) (a) (b) Figure : The graphs of Question (c) Figure : The parabola of Question 7 Page 5
6. A. (f g) (x) = 8x x + 0; (g f) (x) = x + 6x 5 B. (f g)(x) = x; (g f) (x) = x C. (f g) (x) = x + ; (g f) (x) = x + 7. A. y = (x + ) + 5 B. Domain is (, ), Range is (,5] C. See Figure 8. (a) A. {±, ±, ±} B. {±, ±, ±, ±6} C. {±, ±, ± } (b) A. (x + )(x ) B. (x + )( x)(x ) C. (x + ) (x + )(x ) (c) A. x =, x = B. x =, x =, x = C. x =, x =, x = For the remaining questions see Figure A B C Figure : The graphs in Question 8 9. By the Remainder Theorem the answer is. 0. The first four graphs are shown in Figure. The fifth is shown in Figure 5. Page 6
A B C D Figure : The first four graphs of Question 0 Page 7
E Figure 5: The fifth graph of Question 0. A. (, ] [,] [, ) B. (, 7 ] 5 C. (,0) [,]. A. 6 B. + 8 L. 5. A. 6 B. 7 6 C. D. C. E. 8 F. G. H..76 I. 6 J. K. 0. A. x = B. x = 0 C. x =, x =, x =, x = 5 D. x =, x =, x = 5. To prove these identities, use algebra and the basic identities csc θ = sin θ sec θ = cos θ cot θ = tan θ tan θ = sinθ cos θ cos θ + sin θ = 6. (a) A. B. C. (b) A. B. C. D. (c) A. B. C. 6 D. D. Page 8
The graphs are as follows: 7 8 8 5 9 5 8 8 - (a) 5 0 - (b) Page 9
6 5 7 6 6 - (c) (d).5.5 5 NEA, TK May 0 Page 0