Enrichment, Extension, and Applications

Question 1

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Exercises

  1. Sketch the graphs of each of the following functions for \( -2 \pi \le x \le 2 \pi \).
    1. \( f(x) = \sin (x) + \left\lvert \sin (x) \right\rvert \).
    2. \( f(x) = \sin (x) \left\lvert \sin (x) \right\rvert \)
    3. \( f(x) = \dfrac{\sin(x)}{\left\lvert \sin(x) \right\rvert} \), for \( \left\lvert \sin(x) \right\rvert \ne 0. \)
  2. If \( f(x) = x^2 - 6x + 5 \), sketch a graph of the region defined by the intersection of the inequalities \begin{align*} f(x) + f(y) &\leq 0 \\ f(x) - f(y) &\geq 0 \end{align*}
  3. Define \( f(x) = \log_2(x) \) and \( g(x) = 2x-2 \). Find the value of \( x \) such that \( f ( f (g (x) ) ) = 2 \).
  4. Definition: A function, \( f(x) \), is an even function if \( f(-x) = f(x) \) for all values of \( x \). A function, \( f(x) \), is an odd function if \( f(-x) = - f(x) \) for all values of \( x \).

    Prove that

    1. The sum of two even functions is an even function.
    2. The sum of two odd functions is an odd function.
    3. The sum of an even function and an odd function is neither even nor odd (unless one function is zero for all \( x \)).
    4. The product of two even functions is an even function.
    5. The product of two odd functions is an even function.
    6. The product of an even function and an odd function is an odd function.
  5. Define \( f(x) = \left\lvert x-3 \right\rvert \) and define \( g(x) = 6f(x)^2 -4 f(x)+ 3 \).
    1. Determine all values of \( x \) that minimize \( g(x) \).
    2. Determine the minimum value of \( g(x) \).
  6. Find all real values of \( x \) such that \( 0 \lt \dfrac{x^2-11}{x+1} \lt 7 \).
  7. Define \( f (x) = 2^{kx} + 9 \), where \( k \) is a real number. If \( \dfrac{f(3)}{f(6)} = \dfrac{1}{3} \), determine the value of \( f(9)-f(3) \).
  8. Define \( f(x) = x^3 - 1 \) and \( g(x) = x^3 + 1 \).
    1. Prove that \( \dfrac{f(x)}{g(x+1)} = \dfrac{x-1}{x+2} \) for \( x \gt -2 \).
    2. Define \( h( n) = \dfrac{f(2) f(3) f(4) \ldots f(n)}{g(2) g(3) g(4) \ldots g(n)} \) where \( n \) is a positive integer greater than \( 1 \). Find the limiting value of \( h(n) \) as \( n \) goes to infinity.
  9. Define \( f(x) = \left\lvert x \right\rvert \), and let \[ g(x) = f(x - 1) + f(x - 3) + f(x - 5) + \ldots + f(x - 23) \] Note that the sum on the right side contains \( 12 \) terms with the \( k^{th} \) term being \( f\big(x - (2k - 1) \big) \), for \( k = 1, 2, 3, \ldots, 12 \).
    1. Find the minimum value of \( g(x) \).
    2. Find all values of \( x \) that minimize \( g(x) \).
  10. Define \( f(x) = \sin^6(x) + \cos^6(x) \), and \( g(x) = \sin^4(x) +\cos^4(x) \), and let \( h(x) = f(x) + k g(x) \) for some real number \( k \).
    1. Determine all real numbers, \( k \), for which \( h(x) \) is constant for all values of \( x \).
    2. Determine all real numbers, \( k \), for which there exists a real number \( c \) such that \( h(c) = 0 \).
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