Function Exercise

Bài tập toán tiếng Anh chương Hàm số. Những từ vựng quan trọng là: defined, undefined, domain, range, increase, decrease, monotone, odd, even, vertex…

1. For which value of $ x $ is the expression $ \dfrac{x-7}{x+2} $ undefined?
2. Which expression is undefined when $ w= 3 $?
   1. $ \dfrac{w-3}{w+1}. $
   2. $ \dfrac{3w}{w^2}. $
   3. $ \dfrac{w+1}{w^2-3w}. $
   4. $ \dfrac{w^2+2w}{5w}. $
3. Find the domain of the function \[ g(x)=\frac{\sqrt{x^2-1}}{x^2-4}. \]
4. The domain of function $ f(x) $ is $ (0, 2) $. Find the domain of $ g(x) = f(x^2). $
5. Functions $ f $ and $ g $ have domain of $ \mathbb{R} $. In addition, the minimums of $ f $ and $ g $ are $ 2 $ and $ 3 $, respectively.
6. Identify true statements.
   1. The minimum of $ f(x) $ + $ g(x) $ is $ 5 $.
   2. The minimum of $ f(g(x)) $ is $ f(3) $.
   3. The minimum of $ f(g(x)) $ is $ 2 $.
   4. $ f(x)g(x) \geqslant 6 $, for all $ x\in \mathbb{R}. $
7. Find the domain and range of function $ y =\sqrt{-x^2-6x-5}. $
8. What are the domain and range of the function $ f(x)=\dfrac{1}{\sqrt{4-x}} $?
9. State the domain, range and possible symmetries of the following functions
   1. $ f(x)=\sqrt{x^2+1} $
   2. $ f(x)=\sqrt{x+1} $
   3. $ f(x)=\frac{x+1}{x-1} $
10. Shew, without using a calculator, that $ 6-\sqrt{35}
    <\frac{1}{10} $.
11. Function $ f(x) = ax^3 + bx – 1 $, where $ a $ and $ b $ are constants. In addition, $ f(2) = 3. $ Find the value of $ f(-2). $
12. Suppose $ f(x) $ is an arbitrary function on $ \mathbb{R} $. Is function $ g(x) = f(x^2) $ odd or even?
13. Odd functions $ f(x) $ and $ g(x) $ share the same domain. Is function $ h(x) = f(x)g(x) $ odd, even, or neither?
14. Suppose $ f(x) $ is an even function on $ \mathbb{R} $ and it is decreasing on $ (0;+\infty) $. Compare the values of $ f(1.4), f(1.5) $, and $ f(-\sqrt{2}). $
15. For each parabola below, find the vertex, opening direction, and axis of symmetry.
    1. $ y=x^2+2x+1. $
    2. $ y=2(x-1)(x+2) $.
    3. $ y=-2(x-1)^2+2. $
16. Parabola $ y = x^2 + bx+ c $ is symmetric with respect to line $ x = 5 $. Find the value of $ b $.
17. Consider parabola $ y = a(x – h)^2 + k $, where $ h $ and $ k $ are some constants. State the necessary and suffcient condition for each property below.
    1. It opens up.
    2. It does not intersect the $ x $ axis.
    3. It intersects the $ x $ axis at only one point
18. Identity true statements about quadratic function $ y = ax^2 + bx + c $, where $ a \ne 0 $.
    1. It reaches either maximum or minimum at the vertex.
    2. Its domain consists of an increasing and a decreasing interval.
    3. Its whole graph can be in a single quadrant.
    4. It has only one axis of symmetry.
    5. It crosses the $ y $ axis at only one point.
19. A quadratic function $ y = f(x) $ satisfes $ f(4) = f(5) $. Find its axis of symmetry.
20. Write a quadratic function that satisfes each condition. \begin{enumerate}
21. It passes three points $ (0, 4), (1, 9) $, and $ (-1, 3) $.
22. Its vertex is at $ (2, 1) $. It passes point $ (1, 2) $.
23. Write the new equation after each operation on parabola $ y = 3x^2 + 4x + 5 $.
    1. Shift it horizontally by $ -2 $ units.
    2. Shift it vertically by $ 2 $ units.
24. Find the monotonic intervals and range of each quadratic function.
    1. $ y = (x – h)^2 + k. $
    2. $ y = -x^2 + bx + c $
    3. $ f(x) = -(x + 5)(x – 3). $
25. Function $ f(x) = x^2 – 4x + 3 $. Find its maximum and minimum on the interval $ [1;3]. $
26. Given any three points, is there always a parabola that passes them.
27. Quadratic function $ y = f(x) $ is symmetric with respect to line $ x = 3 $. In addition, $ f(4) = 0 $. Find another zero of $ f(x) $.
28. Find the sum and product of the two roots in quadratic equation $ 2x^2 + 13x – 31 = 0 $.
29. Equation $ x^2 – 3x + m = 0 $ has a root of $ -1 $. Find the value of $ m $.
30. Find the maximum area of a rectangle that is inside the triangle formed by the two axes and line $ y = 2 – x $.
31. Function $ s = 600 t-4t^2 $ is the distance of a landing aircraft runs on the runway before a full stop, where $ t $ is time in seconds on the runway. How much time does it take for the plane to stop?
32. Point $ P (0, 2) $ is not on line $ l: y = 0 $. $ A(x, y) $ is a moving point.
    1. Express the distance between point $ A $ and $ P $.
    2. Express the distance between point $ A $ and line $ l $.
    3. Write an equation for the trajectory of all $ A $ whose distance to $ P $ is equal to its distance to line $ l $. What type of equation is this?
33. Which properties best describe the coordinate graph of two distinct parallel lines?
    1. different slopes and same intercepts.
    2. same slopes and different intercepts.
    3. same slopes and same intercepts.
    4. different slopes and different intercepts.
34. Which equation represents a line that is parallel to the line whose equation is $ 2x+3y=12 $?
    1. $ 6y+4x=2. $
    2. $ 6y-4x=2. $
    3. $ 4x-6y=2. $
    4. $ 6x+4y=-2. $
35. If two lines are parallel and the slope of one of the lines is $ m $, what is the product of their slopes?
    1. $ 0. $
    2. $ 2m. $
    3. $ m^2. $
    4. $ 1. $
36. Line $ p $ and line $ c $ lie on a coordinate plane and have equal slopes. Neither line crosses the second or third quadrant. Lines $ p $ and $ c $ must
    1. form an angle of $ 45^\circ $.
    2. be vertical.
    3. be perpendicular.
    4. be horizontal.
37. Which equation represents a line that is perpendicular to the line whose equation is $ -2y=3x+7. $
    1. $ y=-\frac{3}{2} x-3. $
    2. $ 2y=3x-3. $
    3. $ y=\frac{2}{3} x-3. $
    4. $ y=x+7. $
38. Which line is perpendicular to the line whose equation is $ 5y+6 =-3x $?
    1. $ y=-\frac{5}{3} x+7. $
    2. $ y=-\frac{3}{5} x+7. $
    3. $ y=\frac{3}{5} x+7. $
    4. $ y=\frac{5}{3} x+7. $
39. Write an equation of a line that is perpendicular to the line$ y =\frac{2}{3}x +5 $ and that passes through the point $ (0,4) $.
40. Shanaya graphed the line represented by the equation $ y = x- 6 $. Write an equation for a line that is parallel to the given line. Write an equation for a line that is perpendicular to the given line. Write an equation for a line that is identical to the given line but has different coefficients.
41. Which statement describes the lines whose equations are $ y=\frac{1}{3} x+12 $ and $ 6y=2x+6 $?
    1. They are perpendicular to each other.
    2. They intersect each other.
    3. They are parallel to each other.
    4. They are segments.
42. The graph of the equation $ x + 3y = 6 $ intersects the $ y- $axis at the point whose coordinates are
    1. $ (0;2) $.
    2. $ (0;6) $.
    3. $ (0;18) $.
    4. $ (6;0) $.
43. If point $ (-1;0) $ is on the line whose equation is $ y= 2x+b $, what is the value of $ b? $
44. In the graph of $ y<-x $ , which quadrant is completely shaded?
45. Find the equation of the parabola that has vertex $ V = (4,-1) $ and goes through the point $ (0,-2) $.
46. The parabola with equation $ y = 10(x + 2)(x-5) $ intersects the $ x $-axis at points $ P $ and $ Q $. What is the length of line segment $ PQ $?
47. Parabola $ y = x^2 + bx – b $ passes a fixed point regardless of the value of $ b $. Find the point.
48. The vertex of $ y = x^2 + 2x + c $ is on the $ x $ axis. Find the value of $ c $.
49. Point $ (a, b) $ is in the third quadrant. In which quadrant is the vertex of parabola $ y = ax^2 + bx $?
50. Parabola $ y = ax^2 + bx + c $ is in 1st, 3rd, and 4th quadrant but not the 2nd quadrant. Which quadrant is its vertex in? Does the parabola open up or down?