Algebra Practice Test University of South Carolina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Rationalize the denominator: 14 . 3+ 2 2. Given g(x) = 5x2 2x + 3 and h(x) = 9x + 7, find the value of (g h)(2). 3. Find the sum or difference as indicated, and write your answer in simplified form. x + 2a 3 x + 6 x+1 2x 4. Factor and reduce to simplest form: \u0010 5. Simplify the following completely: 6x2 + 11xy 10y 2 . 3x2 + 10xy 8y 2 2 x4 y 5 2 \u0011 34 5 x 3 y 6 . 6. Perform and simplify your answer: \u0010 the indicated \u0011 \u0010 operation \u0011 5 7 5 4 2 3 3ab 16a b 8b 2ab . 7. Solve the following equation: 5(x 7)2 13(x 7) 6 = 0, then find the sum of the roots. 8. Solve for a: 19 3a + 5 = a. and find the sum of all solutions. 9. A radioactive element decays according to the model y = 80e0.21t , where t is in hours. Use natural logarithms to find the half-life of the element. 10. Find \u001a the point (x, y) that satisfies both equations. 2x + 4y = 12 3x 5y = 3 What is the value of x + y? 11. Two investments are made totaling $10,000. In one year these investments yield $650 in simple interest. Part of the $10,000 is invested at 5 12 %, and the rest at 6 34 %. How much more money is invested at 6 43 %? 12. y = f (x) is a linear equation. If f (3) = 5 and f (6) = 9, find the slope of the line. 13. Write the equation (in slope-intercept form) of the line with slope 5 which passes through the point (2, 3). 14. Find the vertex of the function f (x) = (7p)x2 + (28p)x + (11p). 15. Beginning with f (x) = x, find the function g(x) that shows f (x) shifted to the left 2 units, reflected about the x axis, and then shifted up 7 units. 16. Given these three equations, select the true statement below. a. 7(x + 1)3 6= (7x + 7)3 b. 1 1 = x8 8 3x 3 c. (x + 8)2 = x2 + 64 17. Find the domain of f (x) = ( x+8 18. f (x) = x4 x+2 . x5 for x 3 for x < 3 Find f (0). 19. Find the x-intercept(s) of the function f (x) = 6x2 7x 5 , if any exist. 4x2 12x 7 20. Solve for x using the quadratic formula: x2 6x + 2 = 0. Simplify as much as possible. 21. Solve for x: log2 (x + 2) + log2 (x + 6) = 5. 22. Simplify the following expression: ln (ex+2 ) log7 (1) . log4 (256) 23. Given logk (3) = 7, logk (4) = 13, and logk (5) = 22, find logk 24. Solve for x: 8 ln 1 x 2 \u0001 \u0001 12 2 . 5k3 = 72. 25. A real estate developer opens a new subdivision with a certain number of houses. Each year, 12 new houses are built in the subdivision. The number of houses in the subdivision is a linear function of the years since the subdivision opened. Seven years after opening, there were 109 houses. Eleven years after opening, there were 157 houses. How many houses were there at the beginning? What are the units of the slope of the line? 26. Write the exponential function which represents the data in this table. x 2 1 0 1 2 3 4 f (x) 0.0125 0.05 0.2 0.8 3.2 12.8 51.2