Functions #7 - Questions and Answers

Question 1 Write using function notation: The function f evaluated at 3 is negative 5: ________ Answer 1 The function f evaluated at x can be written as f(x). Hence, we have: f(3) = -5 Answer 2 The given statement can be written using function notation as follows: The function f evaluated at 3 is -5: f(3) = -5 Here, f(3) represents the value of the function f when the input is 3, and -5 represents the corresponding output. Answer 3 To write the given statement using function notation, we can represent the function as f(x), where x is the input value. The function f evaluated at 3 can be denoted as f(3). Since the value of f(3) is negative 5, we can express it as: f(3) = -5 In this equation, the left-hand side represents the function f evaluated at 3, and the right- hand side represents the value it equals, which is negative 5. By using function notation, we can clearly indicate the input value and the corresponding output value of the function. Question 2 Plot a complex number graph. Specify the absolute value of the number. 4i Answer Step 1 The given expression is 4i. The Cartesian point in complex plane is (0,4). The graph is shown below: [Image of a graph with y axis labeled imaginary, x axis labeled real, the point (0,4) is plotted] Step 2 The distance formula for points on the Cartesian plane: d = √((x₂-x₁)² + (y₂-y₁)²) d = √((0-0)² + (0-4)²) = √(0 + 16) = √16 = 4 The absolute value of the number 4i is 4. Question 3 How many times can a horizontal line intersect the graph of a function that is one-to-one? Answer 1 Suppose f is an one-to-one function. If (a,b) lies in the graph of f, then there does not exists any x with x!=a such that (x,b) also lies in the graph of f (because f is one-to-one). This means if f is one-to-one, then a horizontal line intersect the graph of f at most once. Thus a horizontal line intersect the graph of an one-to-one function at most once. Answer 2 To solve the problem, let's consider a function f(x) that is one-to-one. In order to find how many times a horizontal line intersects the graph of this function, we need to analyze the behavior of the function. Since f(x) is one-to-one, it means that for every x₁ and x₂ in the domain of f(x), if x₁ ≠ x₂, then f(x₁) ≠ f(x₂). In other words, the function assigns a unique value to each input. Now, let's focus on the horizontal line intersecting the graph of f(x). We can represent this line as y = c, where c is a constant. To find the points of intersection, we need to solve the equation f(x) = c. This means we are looking for values of x that satisfy f(x) = c. If f(x) = c has a solution, it means that there exists an x such that f(x) is equal to the constant c. Since f(x) is one-to-one, there can be at most one solution for x. Therefore, a horizontal line can intersect the graph of a one-to-one function at most once. Answer 3 The number of intersections of a horizontal line with the graph of a one-to-one function can be represented by the equation: n = 0 or n = 1 where n represents the number of intersections. Note: that in this context, n can either be 0 or 1, indicating that a horizontal line can intersect the graph of a one-to-one function either zero times or exactly once. Answer 4 Step 1: Let f(x) be a one-to-one function. By definition, this means that for every x₁ and x₂ in the domain of the function, if f(x₁) = f(x₂), then x₁ = x₂. In other words, no two distinct inputs can yield the same output. Now, let's analyze the intersection of the graph of f(x) with a horizontal line at a specific value, c. This intersection occurs when there exists an x such that f(x) = c. To find the number of intersections, we can set up the equation f(x) = c and solve for x. The number of solutions will give us the number of times the horizontal line intersects the graph. Let's express this equation using LaTeX markup: f(x) = c Step 2: Now, depending on the specific function f(x), the number of solutions to this equation can vary. It is possible to have zero, one, or multiple solutions. If there are no solutions, the horizontal line does not intersect the graph of f(x). If there is one solution, the horizontal line intersects the graph at a single point. If there are multiple solutions, the horizontal line intersects the graph at multiple points. In conclusion, the number of times a horizontal line can intersect the graph of a one-to- one function can be zero, one, or multiple times, depending on the specific function and the value of the horizontal line. Question 4 Find the derivative by the limit process. f(x) = -10x Answer f'(x) = lim(Δx→0) (-10(x + Δx) - (-10x))/Δx f'(x) = lim(Δx→0) (-10x - 10Δx + 10x)/Δx f'(x) = lim(Δx→0) (-10Δx)/Δx f'(x) = lim(Δx→0) -10 f'(x) = -10 Question 5 Name the function f(x) = x² Answer The function f(x) = x² is called a squaring function. Its range is the set of positive real numbers, and its domain is the set of real numbers.