Math 112 Final Exam PRACTICE TEST For the pair of functions, find the indicated sum, difference, product, or quotient. 1) Find (f - g)(-1) when f(x) = -3x2 + 6 and g(x) = x + 5. A) -1 B) 9 C) -3 Find the zero of f. 7) f(x) = 3x + 6 A) -2 D) -2 2) The population of a particular country was 21 million in 1983; in 1990, it was 25 million. The exponential growth function A =21ekt describes A) 3 C) the population of this country t years after 1983. Use the fact that 7 years after 1983 the population increased by 4 million to find k to three decimal places. A) 0.895 B) 0.025 C) 0.035 D) 0.198 B) -3 1 256 A) 3) An initial investment of $480 is appreciated for 9 years in an account that earns 13% interest, compounded quarterly. Find the amount of money in the account at the end of the period. Use the r nt formula A = P(1+ ) or A=Pert n D) 1 16 B) C) D) 10) P(x) = -3x4 - x3 + x2 - 4x + 3 A) B) C) D) Find the requested function value. 11) Find (g(f(4)) when f(x) = -4x + 5 and g(x) = 7x2 - 8x - 1. B) $1441.94 D) $1470.26 A) -311 C) 25 Solve the problem. 4) Find out how long it takes a $2800 investment to double if it is invested at 8% compounded semiannually. Round to the nearest tenth of a year. r nt Use the formula A = P 1 + . n B) 10 D) 934 12) Find (g (f(-2)) when f(x) = x-2 and 2 g(x) = 2x + 1. A) 6 B) 8.8 years D) 9.2 years B) -6 C) - 5 2 D) -3 Write the standard form of the equation of the circle with the given center and radius. 13) (0, -7); 9 A) (x + 7)2 + y2 = 81 B) (x - 7)2 + y2 = 81 Solve the system. 5) x2 + y2 = 25 x -y =1 A) (3, -4), (4, -3) C) (-3, -4), (4, 3) D) -6 Find the correct end behavior diagram for the given polynomial function. 9) P(x) = 5x3 + 6x2 - 7x + 7 Solve. A) 8.6 years C) 9 years C) 2 Solve the equation. 1 8) log4 =x 64 Solve. A) $1038.04 C) $1518.04 B) 6 C) x2 + (y + 7)2 = 81 D) x2 + (y - 7)2 = 9 Find the center and the radius of the circle. 14) x2 + y2 - 14x + 16y = -88 B) (-3, 4), (-4, 3) D) (3, 4), (4, 3) A) (-7, 8); r = 25 C) (7, -8); r = 5 State the domain of the rational function. 3x - 4 6) f(x) = 2x + 12 B) (8, -7); r = 25 D) (-8, 7); r = 5 Solve the polynomial inequality. 15) x2 - 10x + 21 > 0 A) (-∞, -6) ∪ (-6, ∞) B) (-∞, ∞) C) (-∞, -12) ∪ (-12, ∞) D) (-∞, 6) ∪ (6, ∞) A) (7, ∞) C) (-∞, 3) 1 B) (3, 7) D) (-∞, 3) ∪ (7, ∞) Find the horizontal asymptote, if any, of the rational function. x2 + 9x - 6 16) f(x) = x-6 A) y = 1 C) y = 0 23) f(x) = x - 3, for x > 0 2, for x ≤ 0 A) B) None D) y = -9 6 y 4 8x3 - 3x - 7 17) f(x) = 5x3 - 2x + 8 A) y = C) y = 2 3 2 B) None 8 5 D) y = 0 -6 -4 -2 2 4 6 x 2 4 6 x 2 4 6 x 2 4 6 x -2 -4 -6 B) Find the inverse of the one-to-one function. 18) f(x) = (x + 3)3 A) f-1 (x) = C) f-1 (x) = 3 x+3 x-3 B) f-1 (x) = D) f-1 (x) = 6 3 3 x - 27 4 x-3 2 -6 Solve the exponential equation. 19) 3 x = 18 -4 -2 -2 Give your answer as a decimal rounded to the nearest thousandth. A) 6.000 B) 2.631 C) 0.380 D) 1.792 -4 -6 C) 6 Find the zeros of the polynomial function and state the multiplicity of each. 20) f(x) = -5x2 (x - 8)(x + 3)3 2 -6 -4 -2 -2 -4 -6 D) 6 Find the logarithm using the change-of-base formula. 21) log7 9.72 y 4 B) 0.8557 D) 0.9877 2 -6 Find the range of the given function. 22) f(x) = -3x2 - 12x - 17 A) [5, ∞) C) (-∞, -5] y 4 A) -3, multiplicity 3; 8, multiplicity 1 B) -3, multiplicity 3; 0, multiplicity 2; 8, multiplicity 1 C) -3, multiplicity 3; 0, multiplicity 2; 3, multiplicity 1; 8, multiplicity 1 D) -3, multiplicity 1; 3, multiplicity 1; 8, multiplicity 1 A) 1.1687 C) 1.3886 y -4 -2 -2 -4 B) [2, ∞) D) (-∞, -2] -6 Graph the function. Use substitution to determine whether the given number is a zero of the given polynomial. 24) -2; P(x) = x4 + 3x2 - 4 A) Yes 2 B) No Solve the logarithmic equation. 25) log x = -4 6 1 A) 4096 C) 4096 Answer the question. 29) How can the graph of f(x) = B) 1296 D) obtained from the graph of y = x2 ? 1 1296 A) Shift it horizontally 11 units to the right. Stretch it vertically by a factor of 2. Shift it 8 units up. B) Shift it horizontally 11 units to the right. 1 Shrink it vertically by a factor of . Shift it 8 2 Find the domain and range of the function represented in the graph. 26) 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 y units down. C) Shift it horizontally 11 units to the left. 1 Shrink it vertically by a factor of . Shift it 8 2 units down. D) Shift it horizontally 11 units to the left. Shrink it vertically by a factor of 2. Shift it 8 units down. 1 2 3 4 5 6 x Give all possible rational zeros for the polynomial. 30) P(x) = -2x4 + 2x3 + 3x2 + 18 A) Domain: (-2, 1]; Range: [0, 3) B) Domain: [-2, 1); Range: [-2, 5) C) Domain: [-2, 1]; Range: [0, 3] D) Domain: (-1, 2]; Range: [1, 3) Find the requested value. 27) 2x + 1, if x < 1 f(8) for f(x) = 8x, if 8 ≤ x ≤ 12 8 - 2x, if x > 12 A) 64 B) -8 C) 3 1 (x + 11)2 - 8 be 2 A) ±1, ±2, ±1/2, ±1/3, ±1/6, ±1/9, ±1/18 B) ±1, ±1/2, ±2, ±3, ±6, ±9, ±18 C) ±1, ±1/2, ±2, ±3, ±3/2, ±6, ±9, ±9/2, ±18 D) ±1, ±2, ±3, ±6, ±9, ±18 Evaluate the function for the given values of a and b. Then use the intermediate value theorem to determine which of the statements below is true. 31) a = -2 and b = -1 D) 25 P(x) = 7x5 - 3x3 + 4x2 + 5 A) P(-2 ) and P(-1) have the same sign, therefore the intermediate value theorem cannot be used to determine whether P has a real zero between -2 and -1. B) P(-2 ) and P(-1) have the same sign, therefore the function P has a real zero between -2 and -1. C) P(-2 ) and P(-1) have opposite signs, therefore the function P has a real zero between -2 and -1. D) P(-2 ) and P(-1) have opposite signs, therefore the function P does not have a real zero between -2 and -1. Find the requested polynomial. 28) Find a polynomial function of degree 3 with -1, 3, 4 as zeros. A) f(x) = 3x3 + 6x2 + 5x - 12 B) f(x) = x3 + 8x2 + 5x - 3 C) f(x) = x3 - 6x2 + 5x + 12 D) f(x) = x3 + 6x2 + 5x - 12 Find the oblique asymptote, if any, of the rational function. x2 - 6x + 7 32) f(x) = x+6 A) x = y + 6 C) None 3 B) y = x + 13 D) y = x - 12 Find the inverse of the function. 1 33) f(x) = x + 3 8 39) 8x + 5y > -2 A) 10 A) f-1 (x) = 8x - 24 1 B) f-1 (x) = x + 8 3 C) f-1 (x) = 8x + 24 D) f-1 (x) = 1 x-3 8 -10 Find the vertical asymptote(s) of the graph of the given function. x+6 34) g(x) = x-7 A) y = 7 C) x = -7 10 -10 B) B) x = -6 D) x = 7 10 Determine whether the given function is one-to-one. If it is one-to-one, find a formula for the inverse. 35) f(x) = 7x3 + 8 -10 3 x -8 A) f-1 (x) = 7 C) f-1 (x) = 3 x-8 7 10 3 x+8 B) f-1 (x) = 7 -10 D) Not one-to-one C) 10 Provide the requested response. 36) Suppose that a polynomial function of degree 4 with rational coefficients has -4, -3, -3 - i as zeros. Find the other zero. A) 3 + i B) 3 - i C) -3 - i D) -3 + i -10 Solve the system of equations using matrices. Use Gauss-Jordan elimination. 37) 4x - y - 2z = -7 -4x + 8y + 5z = 46 8x - 3y + z = 13 A) {(2, 3, 6)} B) {(-2, 3, 4)} C) Infinite Solutions D) No Solution 10 -10 D) 10 -10 Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. 38) x + y + z = 7 x - y + 2z = 7 2x + 3z = 14 A) {(1, -5, 11)} B) {(2,, 3, 2)} C) No Solution D) Infinite Solutions 10 -10 Determine whether there is a maximum or minimum value for the given function, and find that value. 40) f(x) = x2 - 20x + 107 A) Maximum: -7 C) Maximum: 10 Match the inequality with a graph. 4 B) Minimum: 7 D) Minimum: 0 Determine the intervals on which the function is increasing, decreasing, and constant. 41) 5 4 3 2 1 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 Find the vertex of the parabola. 46) f(x) = 3x2 - 6x - 1 A) (1, -4) C) (-4, 1) y 47) 5 log m - log n b b A) log (m 5 - n) b 5m B) log ( ) b n 1 2 3 4 5 x C) log m5 n Use the compound interest formulas A = P 1 + r nt and n A = Pe rt to solve. 48) Find the accumulated value of an investment of $6000 at 7% compounded continuously for 6 years. A) $9231.77 B) $9004.38 C) $9131.77 D) $8520.00 Determine algebraically whether the function is even, odd, or neither even nor odd. 42) f(x) = 9x4 + 6x - 7 B) Odd b D) log m 5 ÷ log n b b A) Increasing on (-3, 1); Decreasing on (-5, -3) and (0, 5); Constant on (1, 2) B) Increasing on (-3, -1); Decreasing on (-5, -2) and (2, 4); Constant on (-1, 2) C) Increasing on (-5, -3) and (2, 5); Decreasing on (-3, 0); Constant on (0, 2) D) Increasing on (-3, 0); Decreasing on (-5, -3) and (2, 5); Constant on (0, 2) A) Neither B) (-1, 4) D) (4, -1) Solve the rational inequality. Express the solution set in interval notation. -x + 9 49) ≥0 x-5 C) Even Use synthetic division to find the quotient and the remainder. 43) (2x 5 - x4 + 3x2 - x + 5) ÷ (x - 1) A) (-∞, 9] C) (5, 9] A) Q(x) = (2x4 + x3 + 4x2 + 3x); R(x) = 8 B) Q(x) = (2x4 + x3 - x2 + 2x + 1); R(x) = 6 B) (-∞, 5) or [9, ∞) D) [5, 9] Determine whether the graph is the graph of an even function, an odd function, or a function that is neither. 50) C) Q(x) = (2x4 - 3x3 - x); R(x) = 6 D) Q(x) = (2x4 + x3 + x2 + 4x + 3); R(x) = 8 y 10 Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. x+2 44) log 3 x5 8 6 4 2 -10 -8 -6 -4 -2 -2 A) log (x + 2) - log x 3 3 B) log (x + 2) + 5 log x 3 3 C) 5 log x - log (x + 2) 3 3 D) log (x + 2) - 5 log x 3 3 4 6 8 10 x -4 -6 -8 -10 A) Odd The nth term of a sequence is given. Find the first 4 terms. 45) a n = 7n - 2 A) 9, 12, 19, 26 C) 5, 16, 19, 30 2 B) 5, 12, 19, 26 D) 7, 12, 19, 26 5 B) Neither C) Even 51) 10 Graph the solution set of the system of inequalities or indicate that the system has no solution. 54) x2 + y2 ≤ 49 y 8 7x + 5y ≤ 35 6 4 A) 2 -10 -8 -6 -4 -2 -2 2 4 6 10 8 10 x y -4 5 -6 -8 -10 A) Odd B) Neither -10 C) Even -5 5 10 x 5 10 x 5 10 x -5 52) y -10 8 B) 6 10 4 y 2 -10 -8 -6 -4 -2 -2 2 4 6 5 8 10 x -4 -6 -10 -5 -8 -5 A) Odd B) Neither C) Even -10 The graph of a quadratic function is given. Determine the function's equation. 53) C) 10 y y 5 10 8 6 4 -10 -5 2 -10 -8 -6 -4 -2-2 2 4 6 8 10 -5 x -4 -10 -6 -8 -10 Is this the graph of a function? Is its inverse a function? 55) A) j(x) = -x2 + 3 B) g(x) = -x2 + 6x + 9 y 8 C) f(x) = -x2 - 6x - 9 D) h(x) = -x2 - 3 6 4 2 -10 -8 -6 -4 -2-2 2 4 6 8 10 x -4 -6 -8 -10 A) yes, no C) no, no 6 B) no, yes D) yes, no Answer Key Testname: 112MC-LEXAMPRACTTESTSU13 1) A 2) B 3) C 4) B 5) C 6) A 7) A 8) B 9) C 10) A 11) D 12) D 13) C 14) C 15) D 16) B 17) C 18) D 19) B 20) B 21) A 22) C 23) C 24) B 25) D 26) A 27) A 28) C 29) C 30) C 31) C 32) D 33) A 34) D 35) C 36) D 37) A 38) D 39) C 40) B 41) D 42) A 43) D 44) D 45) B 46) A 47) C 48) C 49) C 50) C 51) C 52) A 53) D 54) B 55) A 7