Enrichment, Extension, and Application

Question 1

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Exercises

  1. The function \( f(x) \) satisfies the equation \( f(x) = f(x - 1) + f(x + 1) \) for all values of \( x \). Define \( f(1) = 1 \) and \( f(3) = 3 \); then, \( f(2) = 1 + 3 = 4 \). Determine the value of \( f(1867) \).
    1. For \( x \) and \( k \) real numbers, determine the values of of \( k \) for which the graphs of \( f(x) = x^2-4 \) and \( g(x) = 2|x| + k \) do not intersect.
    2. For \( x \) and \( k \) real numbers, determine the values of of \( k \) for which the graphs of \( f(x) = x^2-4 \) and \( g(x) = 2|x| + k \) intersect in exactly two points.
  2. For \( x \) and \( k \) real numbers, for what values of \( k \) will the graphs of \( f(x) = -2\sqrt{x+1} \) and \( g(x) = \sqrt{x-2}+k \) intersect?
    1. The equation \( y = x^2 + 2ax + a \) represents a parabola for all real values of \( a \). Prove that each of these parabolas passes through a common point and determine the coordinates of this point.
    2. The vertices of the parabolas in part a) above lie on a curve. Prove that this curve is itself a parabola whose vertex is the common point found in part a).
    1. A fixed point of a function is an element of the domain that is mapped to itself by the function. That is, if \( f(x) \) is a function and \( f(p) = p \), then \( p \) is a fixed point of the function \( f(x) \). Determine the fixed points, if any, of the following functions (assume \( x \) is an element of the real numbers):
      1. \( f(x) = 4x^2 - x - 6 \)
      2. \( f(x) = \dfrac{x}{1-x} \)
      3. \( f(x) = 2^{x-1} \)
      4. \( f(x) = \left\lvert x \right\rvert - 2 \)
    2. For what values of \( k \) will \( f(x) = x^2 + kx + 4 \) have no fixed points, exactly one fixed point, and exactly two fixed points?
  3. Let \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are real numbers, be a function defined on the real numbers. If \( 2f(x) = f(2x) \) for all values of \( x \), determine the values of \( a\), \(b \), and \( c \).
  4. Floor and Ceiling Functions:

    Define the floor for a real number, \( x \), to be the the largest integer \( \le x \). We write the floor of \( x \) as \( \lfloor x \rfloor \).

    Define the ceiling for a real number, \( x \), to be the smallest integer \( \ge x \). We write the ceiling of \( x \) as \( \lceil x \rceil \).

    1. If \( f(x) = \lfloor x \rfloor \), find \( f(8.6), f(2), f(-2.6) \).
    2. If \( g(x) = \lceil x \rceil \), find \( g(8.6), g(2), g(-2.6) \)
    3. On the same axes, draw the graphs of \( f(x) = \lfloor x \rfloor \), and \( g(x) = \lceil x \rceil \), for \( -5 \leq x \leq 5 \).
    4. Draw the graph of \( g(x) - f(x) \) for \( -5 \leq x \leq 5 \).
  5. Rounding Function To round a decimal expression to \( k \) decimal places, we add \( 5 \times 10^{-(k+1)} \) to the number and then keep the first \( k \) numbers after the decimal point. For example, \( 5.61497 \) rounded to three decimal places becomes \( 5.615 \) (since \( 5.61497 + 0.0005 = 5.61547\), which becomes \( 5.615 \) when rounded to the first three decimal places); \( 8.297 \) rounded to two decimal places becomes \( 8.30 \); \( 12.34712 \) rounded to four decimal places becomes \( 12.3471 \). If \( r_k(x) \) is a function that takes the real number \( x \) and rounds it to \( k \) decimal places, give an expression for \( r_k(x) \) in terms of \( x \) and \( k \). (Hint: Use the floor function defined above).
  6. Indicator Function An Indicator function, \( f(x) \), is a function that takes the value \( 1 \) if some condition on \( x \) is true, and the value \( 0 \) if the condition is not true. For integers \( x \), define an indicator function: \[ f(x) = \begin{cases} 1 & \text{if \( x \) is divisible by \( 2 \)} \\ 0 & \text{if \( x \) is not divisible by \( 2 \)} \end{cases} \] and define another indicator function: \[ g(x) = \begin{cases} 1 & \text{if \( x \) is divisible by \( 3 \)} \\ 0 & \text{if \( x \) is not divisible by \( 3 \)} \end{cases} \]
    1. Find \( f(2), f(9), f(4086), g(2), g(9), g(4086) \).
    2. Let \( h(x) = f(x)g(x) \). Determine whether \( h(x) \) is an indicator function, and, if so, describe in words the condition it indicates.
    3. Let \( k(x) = \left \lfloor \dfrac{f(x) + g(x) + 1}{2} \right \rfloor \). Determine whether \( k(x) \) is an indicator function, and, if so, describe in words the condition it indicates.
  7. Modulo Function The function \( m_k(x) \) has as its domain the non-negative integers. It is defined such that \( m_k(x) \) is the smallest non-negative remainder when \( x \) is divided by the positive integer \( k \). For example, \( m_4(14) = 2; m_{13}(42) = 3 \).
    1. Find \( m_9(101), m_6(720), m_7(55) \).
    2. Give the range of the function \( m_k(x) \).
    3. Give a mathematical expression for \( m_k(x) \) in terms of \( k \) and \( x \). (Hint: Use the Floor Function defined above).
    4. Draw the graph of the function \( m_4(x) \), for \( 0 \le x \le 15 \).
  8. Prime Counting Function Define the function \( \pi(x) \) to be the number of prime numbers less than or equal to \( x \), where \( x \) is a real number. Draw a graph of \( \pi(x) \) for \( 0 \lt x \leq 50 \).
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