MAT 170 - Precalculus EXAM #1 Dr. Firozzaman
PART I – Free response. Show all your work. Each problem is worth 10 points. f (x h) f (x) 1. Find and simplify the difference quotient , h 0 for the function f (x) 2 2x x 2 . h f( x+ h ) - f ( x ) 2 + 2( x + h ) - ( x + h )2 - (2 + 2 x - x 2 ) Solution: = =2 - 2x - h h h
x 2. Find the inverse of the one-to-one function g(x) . 4x 5 x y Solution: y = . Now replace y by x and x by y to get x = . Solving for y we get 4x - 5 4y - 5 5x y= = g-1( x ) , which is the inverse. 4x - 1
2i(1 3i) 3. Algebraically simplify and write your answer in a bi form: . 3 3i 2i (1+ 3 i ) 2 i - 6 2( i - 3)( i + 1) 4 2 Solution: = = = - - i . 3- 3i 3(1 - i ) 3(1 - i )(1 + i ) 3 3
4. Algebraically solve the equation 2x3 9x 2 38x 21 0 given that –7 is a zero of f (x) 2x3 9x 2 38x 21. -7 is a zero, using synthetic division we have remaining (you do synthetic division part) 2x2 - 5 x - 3 = ( x - 3)(2 x + 1) = 0� x - 3, 1/ 2 are the solutions.
5. Given f (x) 4x 2 24x 32 , re-write the function in standard (vertex) form f (x) a(x k) 2 h by completing the square. Solution: f( x )= 2 x3 + 9 x 2 - 38 x - 21 = 4( x - 3) 2 - 4.
6. Algebraically find all the zeros and their multiplicities for the function p(x) x 3 3x 2 4x 12 . Solution: p( x )= x3 - 3 x 2 + 4 x - 12 = ( x - 3)( x 2 + 4) = 0� x = 3, x 2 i
7. A. Given the functions f (x) 6 4x and g(x) x 2 5x 8 find the following: i. (f- g )( x ) = 6 - 4 x - ( x2 + 5 x - 8) = 14 - x 2 - 9 x
MAT170 – Precalculus EXAM 1 Dr. Firoz ii. ( f° g)( x )= f ( g ( x ) = f ( x2 + 5 x - 8) = 6 - 4( x 2 + 5 x - 8)
B. Suppose h(x) 1 3x , find f (x) and g(x) such that ( f ° g)(x) h(x).
We may choose f( x ) = x and g( x )= 1 - 3 x
PART II – Multiple choice. Chose the response that best completes the statements or answers the question. Write the letter on the blank line. Use capital block letters A, B, C, D, E. Each problem is worth 3 points.
___A____1. Use the leading coefficient test to determine the end behavior of the polynomial function f (x) 8x 2 (x 3)(x 4) 4 .
A. rises to the left and falls to the right B. rises to the left and rises to the right C. falls to the left and falls to the right D. falls to the left and rises to the right E. None of these.
___B____2. Evaluate the piecewise function at the given value of the independent variable. 4x 3 x 2 f (x) Determine f (0). 2x 1 x 2 A. 7 B. 3 C. –1 D. –3 E. None of these.
x ___B____3. Find the domain of the function r(x) . x 2 25
A. (25,) B. (,5) (5,5) (5,) C. (,0) (0,) D. (,) E. None of these.
____C___4. Find the domain of the function f (x) 12 3x .
A. (,4) (4,) B. , 12 12, C. ,4 D. 4, E. None of these.
___A____5. Find the open interval(s) over which the function graphed below is decreasing:
Sp2007 © Department of Mathematics & Statistics – Arizona State University 2 MAT170 – Precalculus EXAM 1 Dr. Firoz
A. , 2 B. , 22, C. 2,2 D. 2, E. None of these. ___C____6. Find the equation(s) o f the vertical asymptotes, if any, of the graph of the rational function x 1 h(x) . 3x 15
A. x = 0 B. x = –5 C. x = 5 D. x = 15 E. None of these.
x 4 2x 2 7 ___D____7. Divide . x 2 15 1 A. x3 2x 2 2x 4 B. x3 2x 2 2x 4 x 2 x 2 1 15 C. x3 2x 2 2x 4 D. x3 2x 2 2x 4 x 2 x 2 E. None of these.
___D____8. A person standing close to the edge on top of 176 foot building throws a baseball vertically upward. The quadratic function s(t) 16t 2 96t 176 models the ball’s height above the ground, s(t), in feet, t seconds after it was thrown. After how many seconds does the ball reach its maximum height? Round to the nearest tenth of a second if necessary.
A. 1 second B. 1.5 seconds C. 2 seconds D. 3 seconds E. None of these.
30(2 3t) ___C____9. The rational function p(t) models the population of foxes that have been 0.045t 1 introduced to Camelback Mt., where t is measured in years. According to the model, what happens to the Camelback Mt. fox population in the long run?
Sp2007 © Department of Mathematics & Statistics – Arizona State University 3 MAT170 – Precalculus EXAM 1 Dr. Firoz A. The population grows to infinity B. The population levels off to 1000. C. The population levels off to 2000. D. The population levels off to 3000. E. None of these.
___C___10. Use the graph to find the value of x such that f (x) 8 .
A. 2 B. 0 C. –2 D. –10 E. None of these.
Sp2007 © Department of Mathematics & Statistics – Arizona State University 4