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Math 141 - Midterm Review Here are some practice problems to get ready for the exam. (1) Chapter 1 (a) 1.1: Functions (i) Write the function f (x) = 2x with domain {1, 2, 3, 4} using dots and arrows, and using pairs of numbers. (ii) Express the domain of (f + g)(x) and (f /g)(x) in terms of the domains of f and g. 1 (iii) If f (x) = x−1 and g(x) = domain of (2f + 2g)(x)? √ x, write a formula for (2f + 3g)(x). What is the (iv) If f (x) = x2 − x, find (and simplify) and expression for the function f (x + h) − f (x) . h (b) 1.2: Graphs of functions (i) Given a function f (x) with domain D (a collection of real numbers), the graph of f consists of all points (x, ), where x is in . (ii) Which of the following curves are graphs of functions? Why? 1 2 (iii) The graph of a function f (x) is pictured below. Is it true that f (2) = 1? Is it true that f (0) = −3? Are there any numbers x so that f (x) = 5? (c) 1.3: Properties of Functions Math 141 3 (i) Describe what feature the graph of a function f (x) must have if it has even symmety. Write the definition of even symmety (that is, write the equation that f (x) must satisfy if it is to be even). Repeat for f (x) odd. (ii) Determine if the following functions are even or odd or neither: x4 , x2 + 1 x3 + 1, ln x, ln |x|, x+ √ 3 x. (d) 1.4 Library of Functions (i) For each of the following functions, what is the function’s domain, range, and graph? Is the function even or odd or neither? x, xn for n odd, xn for n even, c for c a constant, 1 x, √ x, √ 3 x, |x|. (ii) Sketch a graph of the following function. What is its domain? ( −x − 3, f (x) = 1 x, x < −2 −1 < x < 3. (e) 1.5 Graphical Transformations (i) For each graphical operation, describe how to modify the formula for the function f (x) to affect the graph of f (x) in the indicated way: • Shift the graph up by c units. • stretch the graph vertically by a factor of a. • Shift the graph to the right by c units. • Flip the graph horizontally. • Shift the graph to the left by c units. 4 • Stretch the graph horizontally by a factor of a. • Compress the graph vertically by a factor of a. (ii) Sketch a graph of the following function by starting with the graph of using graphical operations: 1 X and −2 − 1. 3x + 2 (iii) Sketch a graph of − ln(−x). (2) Chapter 2: Linear and Quadratic Functions (a) 2.1: Linear Functions (i) What is the general form of a linear function? (ii) If f (x) = mx + b is a linear function, what is the domain and range of f ? Do they depend on m or b? (iii) If f (x) is linear, how many possible solutions are there to an equation f (x) = c, where c is some constant real number? Does such an equation always have a solution? (iv) When is a linear function mx + b even? When is it odd? (v) Is the following data represented by a linear function? If so, what is the function? x f (x) 1 2.5 3 4 5 5.5 7 7 Math 141 5 (b) 2.3 Quadratic functions (i) When if a function of the form ax2 + bx + c not a quadratic function? (ii) If c is a real number and f (x) is quadratic, how many solutions could the equation f (x) = c have? Draw pictures demonstrating each possibility. (iii) Find roots of f (x) = x2 − x − 3 by completing the square. (iv) Find the intersection points of the graphs of f (x) = 12 x2 − 2x − 1 and g(x) = −x2 + 2. (v) What is the range of f (x) = 2x2 − 4x + 5. (c) 2.4 Properties of Quadratic Functions (i) When is a quadratic function ax2 + bx + c even? When is it odd? (ii) What is the minimum value taken by the function g(x) = 3x2 + 2x − 1? (iii) What is the minimum value taken by the function h(x) = −2x2 + 50x? (d) 2.5 Quadratic inequalities (i) Solve −x2 + 3x − 1 ≤ 0. (ii) Solve x2 − x + 2 < 12 x2 + 10. 6 (iii) Is it possible for an inequality of the form ax2 + bx + c > 0 to have (1, 2) ∪ (3, 4) as its set of solutions? How about [−10, 10]? How about {5}? (e) 2.8 Equations and Inequalities involving the Absolute Value (i) Sketch a graph of f (x) = |x + 1| − 1 and use your picture to solve the inequality f (x) > 3. (ii) Solve |x2 − 1| = 1. (iii) Solve |x2 + 6x − 2| = −3.