IB Math AI Topic 2 (Functions) WS 1 PDF

# IB Math AI ## Topic 2 (Functions) WS 1 ### Name: Choe ### Date: ### Period: **1. Write down the domain and range of the following functions.** **(a)** - **Domain**: (-∞, 3) - **Range**: (-∞, 2] **Description of graph**: An increasing line segment from (-3,0) to (3,2). **(b)** - **Domain**...

# IB Math AI ## Topic 2 (Functions) WS 1 ### Name: Choe ### Date: ### Period: **1. Write down the domain and range of the following functions.** **(a)** - **Domain**: (-∞, 3) - **Range**: (-∞, 2] **Description of graph**: An increasing line segment from (-3,0) to (3,2). **(b)** - **Domain**: {-3, -2, -1, 0, 1, 2, 3} - **Range**: {1, 2, 3, 4} **Description of graph**: Discrete points plotted at (-3,1), (-2,2), (-1,3), (0,4), (1,3), (2,2), and (3,1). **2. The following diagrams show the graphs of five functions.** **I** **Description of graph:** A parabola opening upwards with a vertex at (0,1). **II** **Description of graph:** A parabola opening right with a vertex at (-1,-1). **III** **Description of graph:** A straight line with a positive slope, that passes through the y-axis at y = -1. **IV** **Description of graph:** A curved line that flattens towards x = 2 and x = -2. **V** **Description of graph:** A straight line segment starting at (-2,0) and ending at (2,2). **Each of the following sets represents the range of one of the functions of the graphs. Write down which diagram is linked to each set.** **(a)** {y | y ∈ ℝ} **I** **(b)** {2} **II** **(c)** {y | y > 0} **III** **(d)** {1 ≤ y ≤ 2} **IV** **3. The diagrams below include sketches of the graphs of the following equations where a and b are positive integers.** **1.** **Description of graph:** A parabola opening downwards with a vertex at (0,0). **2.** **Description of graph:** A straight line with a positive slope that passes through the y-axis at (0, b). **3.** **Description of graph:** A parabola opening upwards with a vertex at (0,0). **4.** **Description of graph:** A straight line with a negative slope that passes through the y-axis at (0, b). **Complete the table to match each equation to the correct sketch.** | Equation | Sketch | |---|---| | y = ax + b | 2 | | y = ax + h | 4 | | y = x² + ax + b | 3 | | y = x² - ax - b | 1 | **4. The diagrams below are sketches of some of the following functions.** **(i)** y = x² **(ii)** y = x² - a **(iii)** y = a - x² **(iv)** y = a - x **(v)** y = x - a **(a)** **Description of graph**: A straight line with a negative slope. **(b)** **Description of graph:** A parabola opening upwards with a vertex at (0,0). **(c)** **Description of graph:** A parabola opening downwards with a vertex at (0,a). **(d)** **Description of graph:** A straight line with a positive slope. **Complete the table to match each sketch to the correct function.** | Sketch | Function | |---|---| | (a) | (iv) | | (b) | (i) | | (c) | (iii) | | (d) | (v) | **5. The diagram shows a function f, mapping members of set A to members of set B.** **Description of graph**: A function mapping from set A to set B, where A = {-3, -2, -1, 0, 1, 2, 3} and B = {0, 1, 4, 9, 16}. The mapping is defined by: f(-3) = 9, f(-2) = 4, f(-1) = 1, f(0) = 0, f(1) = 1, f(2) = 4, f(3) = 9. **(a)** **(i)** Using set notation, write down all members of the domain of f. {-3, -2, -1, 0, 1, 2, 3} **(ii)** Using set notation, write down all members of the range of f. {0, 1, 4, 9} **(iii)** Write down the equation of the function f. $f(x) = x^2$ The equation of a function g is g(x) = x² + 1. The domain of g is ℝ. **(b)** Write down the range of g. y ≥ 1 or [1, ∞) **6. Let f(x) = x² + 3x + 5 and g(x) = x², for x ∈ ℝ.** **(a)** Find f(3). f(3) = 3^2 + 3(3) + 5 = 23 **(b)** Find (g∘f)(x). (g∘f)(x) = (x² + 3x + 5)² **(c)** Solve (g∘f)(x) = 0. No solution. **7. Consider the functions f(x) = x + 2 and g(x) = 3 - x^3.** **(a)** Solve f(x) = g(x). x = -1.74 or x = 5 **(b)** Write down the interval for the values of x for which f(x) > g(x). (-1.74, 5) **8. Two functions are defined as follows:** $f(x) = \begin{cases} 6 - x & \text{for } 0 ≤ x < 6\\ x - 6 & \text{for } x ≥ 6 \end{cases}$ $g(x) = \frac{1}{2}x$ **(a)** Draw (on the provided graph paper below) the graphs of the functions f and g in the interval 0 ≤ x ≤ 14, 0 ≤ y ≤ 8 using a scale of 1 cm to represent 1 unit on both axes. **Description of graph:** - f(x) is a decreasing line segment from (0,6) to (6,0) and an increasing line segment from (6,0) to (14,8) - g(x) is a straight line passing through the origin with a positive slope. **(b)** **(i)** Mark the intersection points A and B of f(x) and g(x) on the graph. **(ii)** Write down the coordinates of A and B. A(4, 2) and B(12, 6) **(iii)** Write down point N, the x-intercept of the graph of f(x). N(6, 0)